Probably he also knew how to install automated biofilters to the ship's entry points and how to add them to the transporter all along

'
Not sure what the issue is here. Sure, a quadrant is a
huge chunk, but they are significantly farther away than the size of that chunk so at their position it doesn't really matter too much how precise they are - and besides everyone (both in- and out of universe) knows what is meant when they refer to the 'alpha quadrant'.
Suppose we have a movie about a group of time travelers, stranded in, oh, Sri Lanka, 100.000 BC. Bereft of any technology, they only can return to their own time when they reach the spot where the the temporal portal will be built in the future, let's say in Newark. So essentially they have to traverse half the globe using prehistoric technology, which will take them years, if they survive at all. Would it bother you then, if they sometimes colloquially referred to '(north)-America' as their target, rather than 'Newark'?
At least, that would be my response when we're talking about Voyager. For DS9 I agree it is silly to refer to the "alpha quadrant", since even the entire coalition of their forces at the end of the war (Klingons Romulans, Federation) would only make up a tiny fraction of the Alpha Quadrant.