There are most definitely op-amps without controlled limits.
Most general purpose amps do have controlled limits, or they aren't controlled as such, but the worst-case current is still low enough to survive a continuous short, under normal conditions (op-amp mounted to FR-4 board, in still air, etc.).
There are most definitely op-amps without thermal limits. I can't think of one offhand that has a thermal limit. A general purpose amp might be okay into a short circuit for some time, but not under all conditions: high ambient temperature, poor thermal conductivity, unusual circumstances (e.g., mounted on a Teflon board in a vacuum chamber -- basically no thermal conductivity). Rely on that property too much, and you'll screw yourself sooner or later.
Op-amps are not power supply regulators. For that, there are regulators, proper! Regulators almost always have a thermal limit (and whether they have a controlled or gross current limit depends on the design, usually HDO is good vs. LDO sucks).
Tim