The trick to making such an active in-band repeater work would be to ensure that there is adequate isolation from the outside antenna to the inside antenna. Provided the isolation is sufficient, then it won't feedback and oscillate. If there was any coupling inside to outside, then it will find the oscillation frequency very quickly.

The more common approach of providing an external FM antenna system is simple using normal CATV components like amplifiers and splitters. AM you would have to home brew.

If an FM radio lacks an external connector, then you can just couple into the (plastic case?) radio with a length of wire. It is simple to try, so go for it.

Ok. I'll try hanging a length of wire by the radio. Does it just connect to the center conductor of the coax?  Will it matter much what the length is of the wire?  Any chance that this would work for the AM as well?

A couple of feet is good for FM. It's easy to try so try it with minimal effort to start. Quick test.

For AM, you might try a loop to couple into the internal ferrite rod. Many years ago there was a craze for AM band DXing and various products touted the ability to pick-up signals even on the "Alaskan North Slope". Some of these silly plastic case loops provided antenna inputs.

Google AM DXing and you see all sorts of ideas.

It's not likely that you'll have both AM and FM bands on one coaxial cable. You'll probably end up with RG-59 coaxial cable carrying the FM band. And another RG-58 coaxial cable carrying the AM band. It is technically possible to combine them into one cable, but just not very likely to be off-the-shelf.

C Crane makes a nice AM antenna that I've used and liked
called the "Twin Coil Ferrite AM Antenna".
It has a big ferrite that is mounted outside, via coax, and
a small ferrite that is placed near the receiver.