Why hasn't the huge increase in supply provided by Airbnb decreased prices? Simple supply and demand economics would suggest that the huge increase in the supply of rooms to rent by Airbnb should have reduced the price of rental significantly in those cities where Airbnb is most popular. However, this appears not to have happened. Why is that?

There are, for example, ~80,000 airbnb listings in London alone and only ~1500 hotels with ~146,000 hotel rooms.

In fact, it is argued that Airbnb increases prices! 


  Our empirical results show that the overall causal impact of Airbnb on
  property values can be large. For example, in areas within 5km of Los
  Angeles’s central business district the price increase has been 14%.
  Within 2.5km of beaches, the price increase due to Airbnb has been
  almost 10%.


How can this be?
To summarize what's detailed below:


Most research finds some small hotel price decreases following Airbnb entry. But the substitution effect between hotels and Airbnb is hardly 1:1.
Furthermore, some research even found increases in hotel revenues in some areas where Airbnb appeared; this is hypothesized to be due to more discriminatory prices that the hotels can practice because Airbnb customers are generally thought to be more  price sensitive and once they largely leave the hotel market, the hotels are left with less price-sensitive customers, on average.
You are further confusing the substitution/competition of Airbnb with hotels with the effect of Airbnb on long-term rents and/or (residential) property prices. The effect of Airbnb on the latter is indeed generally found to be contrary to that on the hotel market, i.e. because Airbnb offers more (and higher-profit) options to residence owners, an increase in long-term rent prices and property prices is observed in some (but not all) areas where Airbnb operates.




Your research (on hotels) is probably not sensitive enough to small effects:


  There is evidence that Airbnb increases the supply of short-term travel accommodations and slightly lowers prices.  [...]
  
  Airbnb is essentially a positive supply shock to short-term accommodations. Like all positive supply shocks, it should be expected to lower prices. There is some accumulating evidence that Airbnb does exactly this. Zervas, Proserpio, and Byers (2017) examine the effect of Airbnb expansion across cities in Texas. They find that each 10 percent increase in the size of the Airbnb market results in a 0.4 percent decrease in hotel room revenue. They find that most of this revenue decline is driven by price declines. Evidence of the positive supply shock is particularly evident in the 10 American cities where Airbnb’s presence is largest. Dogru, Mody, and Suess (2019) find a negative correlation between Airbnb expansion and hotels’ average daily rates in the 10 U.S. cities with the largest Airbnb presence.


Also, Airbnb and hotels are not necessarily perfect substitutes:


  Further, it is possible that by substituting more strongly for a less-expensive slice of the traditional hotel market—leisure travel as opposed to business travel, for example—that Airbnb introduction might actually be associated with raising measured short-term travel accommodation prices, through a composition effect.


See elasticity of substitution for the theoretical notion.

Also, an MA thesis/study in Norway (not cited in that review) seems to support this idea of hardly perfect substitutability of hotels and Airbnb:


  I find that a 10 % increase in Airbnb supply decreases hotel
  revenue by 0.3 %. This effect is around 20 times smaller than the effect on hotel revenue of an increase in hotel supply. 




As for your last claim "In fact, it is argued that Airbnb increases prices!" that source seems to be talking about property prices:


  Our empirical results show that the overall causal impact of Airbnb on property values can be large. For example, in areas within 5km of Los Angeles’s central business district the price increase has been 14%. Within 2.5km of beaches, the price increase due to Airbnb has been almost 10%. This shows that in areas that are attractive to tourists, increases in property values are substantial, while in areas without much tourist demand (e.g. Pasadena), effects are small.


I don't see any contradiction with the findings about hotels.

Also (long-term) rental prices being increased by Airbnb by offering another (short-term) use for the same (residential) resources is consistent with this increase in property prices. Going back to my first (EPI) link:


  The single biggest potential cost imposed by Airbnb comes in the form of higher housing costs for city residents if enough properties are converted from long-term housing to short-term accommodations. If property owners take dwellings that were available for long-term leases and convert them to short-term Airbnb listings, this increases the supply of short-term rentals (hence driving down their price) but decreases the supply of long-term housing, increasing housing costs for city residents.  [...]
  
  The mirror image of Airbnb’s positive supply shock to short-term travel accommodations is its negative supply shock to long-term housing options.  [...]
  
  Airbnb—though relatively new—is already having a measurable effect on long-term housing supply and prices in some of the major cities where it operates. For example, Merante and Horn (2016) examine the impact of Airbnb on rental prices in Boston. The authors construct a rich data set by combining data on weekly rental listings from online sources and data from Airbnb listings scraped from web pages. They find that each 12 Airbnb listings per census tract leads to an increase in asking rents of 0.4 percent. It is important to note that this is a finding of causation, not just correlation. [...]
  
  Barron, Kung, and Proserpio (2018) undertake a similar exercise with different data. They create a data set that combines Airbnb listings, home prices and rents from the online real estate firm Zillow, and time-varying ZIP code characteristics (like median household income and population) from the American Community Survey (ACS). To account for the fact that rents and Airbnb listings might move together even if there is no causal relationship (for example, if both are driven by the rising popularity of a given city), they construct an instrumental variable to identify the causal effect of rising Airbnb listings on rents. Using this instrument, they find that a 10 percent increase in Airbnb listings in a ZIP code leads to a 0.42 percent increase in ZIP code rental prices and a 0.76 percent increase in house prices. They also find that the increase in rents is larger in ZIP codes with a larger share of nonowner-occupied housing. Finally, like Merante and Horn, they find evidence that Airbnb listings are correlated with a rise in landlords shifting away from long-term and toward short-term rental operations.




If you want results that appear more genuinely paradoxical, there is a 2016 paper with those:


  This paper gives an overview of the activities of Airbnb in 14 European cities. Since
  Airbnb provides an online accommodation platform linking property owners and
  visitors, it could potentially affect both the hotel market and the domestic rental market
  in the localities in which it operates. We discuss the structure and the segmentation of
  the accommodation market, and then present some descriptive statistics on Airbnb
  activities in the 14 cities. Finally, we present some estimates of the impact of Airbnb on
  hotels and on rents, among the first estimates for European markets. We find Airbnb’s
  presence in a market has a negative effect on hotel occupancy rates, but a positive
  effect on total hotel revenue and the average daily rate they charge.  In the two cities we
  consider, the platform’s impact on the rental market is ambiguous, suggesting local
  market conditions are important. [...]
  
  We find that a rise in the number of Airbnb listings in a city was associated over this
  period with a fall in the average hotel occupancy rate, but an increase in the average
  daily rate received by hotels. The combined effect on total hotel revenues was
  ambiguous to slightly positive. On the other hand, the arrival of Airbnb is positively
  correlated with the rental price index in London, but not in Berlin.


However even the seemingly paradoxical divergence between hotel occupancy and their revenues (observed in Europe) has an (elasticity-based) explanation:


  Business travellers are much more able or
  willing to pay for convenience and reliability. For example, Airbnb hosts can cancel
  reservations at short notice and the platform does not penalize the hosts who renege on
  their promises, or compensate the users who find themselves nowhere to stay at short
  notice. The informality of the sharing economy does not (yet) sustain service norms. On
  the other hand, we always expect hotels to observe their promises. By screening out
  budget travellers, hotels may be better able to identify customers with less price-elastic
  demand and thus possibly even raise prices and obtain higher revenue thanks to the
  greater scope for price discrimination. Thus the entry of flexible low cost supply could in
  effect push up the prices and revenues of some hoteliers, though low-cost hotels may
  suffer. 


Relevant theoretical notions for this last para: price elasticity of demand (PED); price discrimination.

Unfortunately, empirically testing this latter hypothesis that customers with lower PED go to hotels, wasn't done in that study. This requires more data on the customers than supply-side statistics convey.

There is however some corroborating independent research that confirms that Airbnb customers in the UK have the same age profile as low-cost/economy hotel customers.


  people
  aged 25–34 years are the largest group of users of
  Airbnb.  [...]
  
  Economy hotels are the most popular among
  people aged 25–34 years. 


(Of course, age is not a perfect proxy for income or elasticity of demand.)