Nothing about utility maximization model says utility function is unbounded

Yep.

the only mathematical assumptions for a well behaved utility function are U'(x) >= 0, U''(x) <= 0

Utility functions aren't necessarily over domains that allow their derivatives to be scalar, or even meaningful (my notional u.f., over 4D world-histories or something similar, sure isn't). Even if one is, or if you're holding fixed all but one (real-valued) of the parameters, this is far too strong a constraint for non-pathological behavior. E.g., most people's (notional) utility is presumably strictly decreasing in the number of times they're hit with a baseball bat, and non-monotonic in the amount of salt on their food.