Replace check.sh and/or ci.sh with tox/nox

[jakkdl]

Tox sounds wonderful. I've been meaning to suggest it. I've actually been working on some reusable workflows for tox-based projects + experimented with some cool GHA integrations.
When you get to it, I may offer some help / experimentation in simplifying things here.

Originally posted by @webknjaz in #3157 (comment)

thread contains more discussion on tox vs nox. Whoever wants to Do It can probably just decide on either.

Initial PR does not need to be a huge one that replaces all of check.sh & ci.sh and overhauls CI stuff, an incremental approach should work perfectly fine here.