Both the Mac mini and the MacBook Air are very much Macintosh computers, just as your old iMac is.
The macOS software has evolved a bit since 10.15 Catalina, but not so much that the operation will be foreign to you. I think either of these computers would serve you well.
The Mac mini is a powerhouse in a small package. You can continue to use the keyboard and mouse or trackpad from the old iMac if you like. You will have to budget for a new monitor. You can find a fine screen in the $150-$350 price range. I think the mini is a great Mac.
The big questions really are:
1) Do you need to have a computer that is mobile and not tethered to a desktop?
2) Is portability worth the extra $ that a MacBook Air will cost? Would you want to travel with it?
You might visit this Apple website and see what you come up with: Mac - Which Mac is best for me? - Apple
Any new Mac, whether the Mac mini or a MacBook Air, will run rings around the old iMac.
In my humble opinion, the benefits of current support, warranty, modern macOs software and increased internet security justify the transition to a newer Mac.