Some of the Mossbergs are stamped US Property... but there are many .22lr trainers.. Most of the milsurp trainers can get pricey, trust me, some of these little rifles hurt when I bought them...the Romanian M1969 is still fairly common to find for...250 or less.
This is the commercial equivalent of the Mossberg M44US, the Mossberg 144LSB, sometimes they can be found for a little less than the real deal M44US.
An Enfield trainer
A German KKW Mauser Trainer
A MAS45 trainer..
A Polish Wz.48 Mosin trainer
Romanian M1969 trainer...in blond and red
A Russian CM2.. don't quite know if it's deemed a trainer.
A fun commercial, if you use the hottest 22lr you can find, it looks and feels like a real M1 Carbine, the German made Iver Johnson, I can even fix a M1 Carbine bayonet on it.
Somebody is making one now in current production, someone like Chiappa... no clue how they are.
Another commercial, the Norinco JW-15, this is what the TU-KKW and TU 33/40 are based off of, they are all copies of a Cz BRNO-1 rimfire. They are pretty easy to find, accurate and cheap.
Another you could find tucked away, not really a trainer, not really a true commercial.. it was deemed more of a garden rifle.. a Cz 468, single shot, but this one is accurate enough, you only need one shot, and it's a Cz.
Many moons ago, maybe about the time or just before MTV came on the air, I cut my teeth on this, an old arse Marlin Model 99, it predates the Model 60, my old man handed me a bullet at a time, till he felt I was ready, that sucks with a tube feed..but I still have it, shoot it and love it and can load it in a heartbeat.
I could keep going, but those are just some ideas for trainers.