I don't know about the fine tuning tailpieces, really.  I have one on my acoustic, carbon/plastic composite with the built in fine tuners.  Not having done A/B style comparison in person, I don't know which sounds better.

I do note though, that the "pro" players don't seem to use them.  If they were actually good for the sound, I reckon they would.  

Logically, any mechanism with moving parts like a fine tuner is going to damp the vibration of the string a bit.  Most of the "nice" violins I've seen have wood tailpieces, and usually only one fine tuner (for the E string).  There are probably reasons, but I don't really understand enough about the instrument yet for them to be obvious to me.  Still too early in the learning curve.

But there are some very pretty wooden tailpieces available, and I really don't like the look of molded plastic on my acoustic violin. LOL

So far as taking parts off, well, those changes are all reversible when it comes to things like the chin rest and shoulder rest.  I found the acoustic violin comfortable without the shoulder rest, so I never used one for it after a brief try to see if maybe I liked it better with a shoulder rest.  With my first acoustic, I found it much more comfortable to play without the chin rest as well.  But everyone is built different, and what one person needs to play comfortably will be different than what another person needs or likes.

I don't really think of it as a purist viewpoint or a tone thing so much as "if you don't need it, don't use it".  If you do need it or find it helpful, then by all means you should use it.

I have heard concerns that playing without the chin rest in particular would restrict a player to "first position" mostly and one couldn't play complex or advanced pieces.  Didn't seem to slow old Paganini down much, though.  While it might take developing or re-learning different techniques than are taught for modern style playing with the chin rest, obviously it is possible to play complex pieces and use the full range of the fingerboard somehow. 

Now, it might not be possible without a chin rest and shoulder rest to satisfy a teacher who is used to the posture and hold that use them.  I don't know on that, I haven't dealt with a teacher for violin or looked into any of the method books yet.  But the way we think of as "proper" to play the violin today is not how it was played in the past (including some of the periods where a good bit of famous violin music was composed and performed) and how it is played in another century or so may be different than how it is done today.  Music is like that, ever evolving and changing, not only in sounds and musical styles but also in playing techniques.  

I paid about $7.50 for the set I put on my electric, and they have worked out pretty well.  But I made it a point to pick ones where the screws/threads were brass rather than plated, and I checked them over to make sure they turned and operated smoothly, and put a little white lithium grease on the threads when i checked them over before installing. 

So I'd say it's possible to get tuners that are ok for a bit less.  But you want to check them over before putting them on, because as a general rule, the cheaper a piece of hardware is, the more inconsistent it's manufacture was.   And sometimes, the cheaper hardware will be the stuff that failed a QC check where the stuff that passed is sold at a higher price.

As an anecdote that doesn't relate to violins, but does relate to a wood product that is partially done by hand and partially done by machine..

My step-father worked for many years at the company where almost all Louisville Slugger baseball bats were made through the 1970s.  They supplied white ash baseball bats for everyone from major league pros to kids.  He explained once that there was actually no difference at all between how they were made.  The pro players would come in and pick their bats out of thousands that were in the plant on any given day.  Some of the pros tapped them, some swung them a bit, some checked how they bounced if dropped on end and that sort of thing and would pick out the ones they wanted.  They all seemed to have different qualities they considered important and were looking for.

But the important thing, in my step-father's opinion, was that they didn't sort any out special for the pros or do anything special unless the pro wanted something done after they'd picked their bats.  Some wanted the finish sanded off and other such things.  They usually signed the bats they picked so they didn't get mixed up with other players' bats.  But most days there were no pros picking out bats, and any kid in any sporting goods store or department had just as good a chance of finding one that was really good as any pro did.  So the difference was all in the selection.