As a single data point: I'm frequently on old stables because I just don't think to check. If I check, and there's an update, it immediately downloads it which ties up rustup for a good 10-15 minutes, so I'm slightly disincentivized from checking.
Also, I'm lazy.
I think a periodic background check followed by a little "Note: update to $TOOLCHAIN is available" would be nice... so long as I can silence it if I need to.