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What reliable fun .22 would you recommend for under $350?"
I'll echo just about everyone else;
My 10/22 ran ~30000 rounds without issue. The only mod was the bolt release done with a die grinder. I cleaned the gun twice, once when I bought it and once at ~20000 rounds. It was an experiment

. I gave the rifle to a new shooter, it's still going strong. You asked for "fun and reliable", I wouldn't list it if you asked for accuracy. You couldn't hit a barn if you were inside it.
My Savage bolt is amazing. It's still lnib

at ~1500 rounds.
I won't claim it to be as accurate as a CZ, but it's in your budget, a CZ isn't. It's accurate enouph for me. I bought the TRR SR, my brother has the $200 version, it shoots the same.
My favorite is my sons Henry lever. For us it's the most fun to shoot, holds 15 LR, very accurate, and it's an action shooter. It too is young, ~1000 rounds, but never an issue.
One of our games is shooting clays at 30 yards. We'll line up rows of 8 or 10 clays, first person to break their row wins. Whoever shoots the Henry, wins!