I'm a GP, and I see a huge range of people - all ages, faiths, classes, varied interests etc. The one thing they all need to do is trust me, otherwise I can't do my job. And in order to trust me they have to be able to relate to me.
If I make a big statement with my appearance (tattoo, power suit, skinhead haircut, skin tight mini dress) there is a danger they may put me in a category that they can't relate to, and then they may not feel able to confide in me.
For that reason I try to make myself a fairly bank canvass at work. I wear plain smart-ish clothes, small amount of understated jewellery, tiny bit of make-up. That way I can be anyone, and therefore people in all walks of life can open up to me, which is - after all - the whole point of my job.
I see teachers in a similar way. I don't want to know too much about what they might be like when they're not at work. And in my opinion a tattoo says more about someone than I want to know, if they are my child's teacher.
OP you said yourself that you liked your midwife all the more because she had tattoos, which goes to show that we all make judgements on appearance, be they positive or negative. I think in some jobs people should try and inspire as little judgement as possible.
That's my view anyway, for those who were curious.