Thanks for the reminder, I should probably update the file... I see it's been almost 2 years.
[minutes later] Whew! It's grown from 27580 to 35943 bytes, or 11505 vs 14623 after GZIP. Uncool.
So yes, I do host it locally, but that has nothing to do with SEO, I just find it pointless to fire an external request for something that I can get to the client faster with HTTP/2. And I can cache it for longer than 2 hours, which is the time that Google seems to be setting. However, I don't think it visibly affects speed, so I wouldn't necessarily recommend hosting it locally. Either way is fine.
Analytics was working just fine before I updated the file, so the changes appear to be backward compatible.
I'm still using analytics.js, though. The gtag.js implementation differs a little, but the
help page [support.google.com] only talks about code updates so I suppose that too should work locally. I see no real incentive to switch, and the gtag.js file size is 2x that of analytics.js.