Why do stock exchanges not operate at a fixed frequency? Would this not be more fair towards traders who don't trade at high frequency?

And would it not be possible to distribute this fixed frequency to other exchanges, modulo relativity?

The reason I'm asking is because I recently learned that high-frequency traders are beating inter-market sweep orders.
As long as trading faster is not explicitly forbidden, there will be people who want to trade faster if they can take advantage of it. 

If you are able to trade fast enough, you can take advantage of arbitrage situations.
For instance, if you can detect a fall in the price of soy on the New-York exchange, buy soy on the New-York Exchange, and sell it on the Chicago exchange before the price in Chicago adapts, you will profit from it. 

So, on the positive side, the answer to your question in the title "Why do stock exchanges not operate at a fixed frequency?" seems to be "because it's not impossible/forbidden to operate faster and people benefit from it".

Now, on the normative side, there are certainly a fair amount of people who believe that stock exchanges would perform ``better" if they operated at a fixed frequency.

Roughly, increasing the frequency of trades is good if it allows prices to adapt faster to new economic information. Some people argue, however, that past a certain threshold, increasing the speed of trade cannot foster more effective price adjustments. The argument is that relevant economic information only arrives every so often, and there is no point in trading faster than the fastest stream of relevant information.
Past this threshold -- so the argument goes, increasing the speed of trade only allows for increased high-frequency speculation in which traders benefits from arbitrage situations that are unrelated with ``actual" economic fundamentals. 

I don't think they are the first to propose it, but Budish, Crampton and John have a recent pair of papers in which they advocate for a cap on the speed of transactions for similar reasons : The High-Frequency Trading Arms Race: Frequent Batch Auctions as a Market Design Response, and Implementation Details for Frequent Batch Auctions: Slowing Down Markets to the Blink of an Eye.