The SE line has been designed to hit a low price point. Typically, it used older designs (the 5 series, iPhone 6-8) which had large volumes to keep component and production costs low.
The Mini didn't have that. It looks like a two-year run with low volumes. If Apple has to make a Mini with no Face ID with a new screen and a Touch ID side button, it has to pay for that work somehow, and I doubt they'll hit their $399 price.
So, I doubt the Mini will be the new SE3. If you look at the combination of people wanting larger phones, along with a phone with cheaper components, and a long production run, I think the iPhone XR (which is still currently being sold, BTW) will turn into the SE 3 in Spring 2023. (an XR with modern internals like 5G, a better camera, and a fast processor)