Why can't the "mask exempted" people wear a face shield?

They can, but most places want you to wear a mask, even if you're exempt.

I have anxiety, one of the conditions that makes you exempt from wearing a mask. I wear one when I have to which includes my workplace. I make my trips out short if I have to go to an indoor public area. 

Face shields are not all that protective when worn alone, they should be worn with a mask. People should be saving their covid pay instead of taking it when they don't want to wear a mask. 

Personally I would not exempt ANYONE from wearing a mask.   Most of them are just pansies.

They do where I work.  When the whole mask thing started, people took the 2 weeks covid pay off from work, mostly because they didn't want to wear the mask.  So then the employer said, okay if you get a note from doctor, you don't need a mask.

THEN.... the employer said, okay reverse what we just said.  You HAVE to wear a mask.  If you can't wear a mask, we the employer will provide face shields that you can wear.  So maybe 10 or 20 people wear a face shield, everyone else wears masks.

they probably dont like putting stuff on their face

They can.  It's never been a law; just common sense,  which obviously isn't that common...