The whole relative effectiveness of bow and firearms is a complicated issue but I think one that was basically answered around 100 years before this time.
The Bashkirs were in almost continuous revolt at this time and barely trusted with anything dangerous when not in revolt. Also I don’t think that they are in the same kind of league as the Tatars. The Tatars were a big power in the area and had only a few years before this era raid far into Russia every year.
I have idea on the Comanche but again I don’t think they are in the same kind of league as the Tatars. They are also, of course, a long way from the area and in a completely different set of circumstances. Any comparison is meaningless.
I am sure that some Tatars would have bows as well. Some people were wandering around Normandy in 1944 with swords, bows and all sorts. Similar the LDF (Dad’s army) were equipped with kitchen knives on broom handles and similar. That doesn’t mean that the standard armament in WW2 wasn’t bolt action rifles, SMG, etc.
As a general rule at this time good troops used firearms. Rubbish troops and troops from isolated areas used bows, until they could get firearms. The Tatars were a large rich power with a good army.