Thank you all for the help!
There are a lot of athletic and sporting goods stores over here, so I thought that finding athletic shirts for this purpose would be fairly easy. It's just as well that I can try them on in the store.
As for the larger-than-needed armpit holes - I've had some experience with those when looking for sleeveless tops, but not much.
Would it be something one could tailor into fitting better?
A larger size could allow you to take in the sides, and with it the holes for your arms. Or, perhaps they could just be taken in alone, without the need for a larger size. The last might work better with a women's shirt, if it's cut to fit one's chest size; then the material wouldn't be as tight as it would when doing the same with a men's shirt if it's spread out in the right places. But then, women's shirts don't usually have that problem.
I think I saw a few guides -somewhere- for making athletic shirts (or rather just spandex shirts, I think) into binders. Has anyone tried that, and - same questions, really: any guides/materials/brands you'd recommend, advantages, disadvantages, how long did they last, and how well did they work?