What gives Turkish coffee that unique "quality"? I'm wondering what makes Turkish coffee made out of original Turkish packed grounded coffee taste so differently (better) from what I can do from beans that I ground myself or in the shop, following basically the same technology? 

I had a few hypothesis:


It's all has to do with right Cezve, i.e. pot. However, buying
'true' Turkish coffee pot didn't help. Actually I get much better
results with Turkish ground coffee comparing to the one sold
by weight even if the latter costs more.
They add cardamom to the coffee. I tried it, coffee tastes better, but
it's still not the same.
They ground it very very finely. I asked the seller to ground my
coffee ad fine as they can but the result is still poor. (Actually this 
is still an open question, because they can use better machines than the ones I have access to.)
They use special coffee mixes.


Then in Wikipedia I read that it is mostly about fineness of the grinding, but burr mill and mortar proposed there didn't work in my case. I also bought turkish coffee mill like this: 


I tried to grind the coffee with this mill many times, results are always poor. Does anybody know what is the secret? Can I make my own coffee that would be the same quality (or better) than, say, Mehmet Efendi?
Assuming that:


you have found the best available beans in your market.
which are then finely grounded 
and used instantly after the grounding (have in mind the finer the grounding the faster the coffee taste deteriorates as it both absorbs humidity and all smells from the environment with it and in the same time releases its aroma). 


You can look for: 


right Cezve : the pot shouldn't be a steel, as the temperature rises very fast which leads to bitter/ burnt taste. The preferred material is copper. 
right preparation methods : make it as slower as possible and wait for it to create a thick cream (caimác) that will protect its aroma as in espresso. To achieve that you can use either direct flame (as you can directly see the flame and control the amount of heat transfered):





or sand (which has large heat capacity and it takes a lot of time to get hot): 




steer it frequently till it boils: to homogenize it perfectly; otherwise the coffee and sugar go down on the bottom and melt/ burn and deteriorate the taste.