Rewiring the motor to 115v is probably a no-go, since it's a 3hp motor (continuous), which I believe is more amps at 120v than could be drawn from a normal circuit.
For safety, I'm wondering what kind of faults wouldn't be protected against. The starter has built-in thermal protection (heaters), which should protect the motor from the unlikely use case where I'm pushing the saw too hard, or more likely, a stall on start-up.
The breaker should still protect against shorts. I'm not knowledgeable enough to know what other faults could lead to an over current fault that one or the other of those wouldn't catch.
Would anybody care to educate me?
Also: the most dangerous aspect of a table saw is probably not the potential for electrical faults :-)
I'm having a hard time coming up with a compelling failure mode, given that thermal protector. Perhaps when the rotor locks, the thermal protector has a slower curve than the saw's wiring (which is assumed to be protected by the circuit breaker's curve). I would hope the saw wiring would be a bit oversized such that it too was protected by the motor thermal protection, but without doing an engineering analysis of the saw itself, you don't really know.
Unknown unknowns are my real point though - you're operating outside of the electrical code. Same as if you wired up a workbench of 5-15 receptacles with 10ga and put a 30A breaker on it - realistically it's going to be fine, but we avoid doing this. All those different NEMA plug sizes exist for a reason. 5-15/5-20 is really an anomaly where you can plug the smaller into the bigger, and even there I've got to wonder if eg old table lamps on a nice stiff 20A circuit are a good idea, or if they just haven't been focused on because they're statistically unimportant (see also: strings of Christmas lights with receptacles at the end).
FWIW I'd probably do the same thing you did, but at the cost of making sure I unplugged the saw whenever I was done using it until I made a proper permanent circuit. But that is not professional advice!
FWIW if the nameplate of the motor specifies current draw at 120V, you can likely reconfigure the motor to use 120V instead of 240V.
[0] take a look at the gauge of a dryer's internal wiring, which is regulated by UL/CE and not NEC.