Just to give some background on this project: it's indeed something I'm doing on my own, independent of Stack Exchange. There are more independent 'volunteer' projects around the Stack Exchange network, e.g. Smoke Detector (which I'm also a member of) which automatically flags spam across the network (including MathOverflow), and a dozen bots performing various moderation tasks on Stack Overflow. Some of these tasks could be done by Stack Exchange the company, but they simply lack the budget and/or (wo)manpower to do so. I'm not asking for permission from every single community, as that would amount to me having to post 173 questions on each single meta site. I've pointed out some means to contact me on the GitHub page, but this Meta post will do fine; thanks @MartinSleziak for notifying me about this post.
You're right that as part of the transition from HTTP and HTTPS, all HTTP links to sites inside the Stack Exchange network were automatically rewritten to HTTPS by a script made by Stack Exchange employees. However, that was a relatively simple script; it did not check whether the target still existed, or could instead be retrieved via the Wayback Machine.
One of the nice things about that script was that it didn't bump old posts; mine does (which is good, so other users can (re)view what it does), but there are other scripts that bump things (recent example). If this community finds 1-3 bumps per 36 hours too annoying, I can decrease the frequency. Other conditions can be discussed; if, for example, the bot should only attempt to fix posts with a positive score, I can try to program those conditions into the bot. Switching it off entirely for MathOverflow is also an option, but to me that would be equivalent to a government ban on restoration of old paintings. For what it's worth, at this moment, 97.5% of the suggested edits by the bot have been approved (network-wide).
For the record, there are currently 41 posts left on MathOverflow which the bot will try to fix.