I think a lot of people just have an aesthetic sensibility that gets bumped by the sight of these things clamped onto headstocks, headstocks whose shapes are very important to some. I don't like to see them, but it's just a visual thing, it doesn't make me feel one way or another about the professionalism of the user.The people that have a problem with clip-on tuners are usually geezers like me because they never saw George Harrison or Eric Clapp or Jiminy Page with one in the 60s and 70s.
BUT! I have no problem with 'em. I have no need to make a fashion statement. I do have a need to play in tune, and I will use all the help I can get.
If you have not tried the USB-rechargeable BOSS clip-on tuner, holy cats...never suffer dead batteries again. These are superb: