You're allowed to have as many **30A** outlets as you want off a 30A breaker. However all the outlets must be 30A (NEMA 6-30 or 14-30). See NEC 210.21(B)(3).
Forgive me if I make a rude assumption, but "string of 4 outlets" strongly implies these are common 120V/15A outlets. Those are only allowed on 15A breakers, or (due to an exception) on 20A breakers if there are 2 or more outlets and the wire is 20A-rated.
Breakers have 2 trip modes. First is thermal trip, which is based on how long and how much overcurrent. So a 3x overcurrent for 1/10 second, or a 150% overcurrent for 5 seconds, the breaker should tolerate that since that won't overheat your wires. Second is magnetic trip, which snaps the breaker if instantaneous current is far above trip rating like 1000%, e.g. 200A for a 20A breaker.
First, I am gobsmacked that a saw would trip a 30A breaker. Like any motor load, a saw has a "spiky" startup load which is called Locked Rotor Amperage (LRA). Too little for a magnetic trip, and it should spin up quickly enough not to cause a thermal trip.
My guess is you already tried changing the breaker (that illegal 30A breaker didn't get there by itself) and that has given no relief.
We know we can't be *too far* into the trip zone, because using the extension cord prevents the trip (the thin #16 wire is limiting LRA). Caution: do not run an extension cord coiled - so many wires are packed next to each other that they can't cool.
Now that points to 2 possibilities. #1 is the saw motor is actually defective. Motors are clever but simple beasts: there is nothing electrical on the spinning parts. It's possible that a careful and searching clean-out of the motor may do the trick. Saws certainly have problems with sawdust buildup, which then gets wet. You also might try changing the capacitor; the capacitor's job is to get the motor turning in the right direction, and if the cap has a problem the motor might be spending too much time at locked-rotor amperage. Try a good cleanout, then the cap, then a teardown.
#2 is that you are unaware of other loads on the circuit; these loads plus the saw are causing the breaker trip.
The usual "unmentioned" is the dust collector. The problem with "locked-rotor amperage" will be much worse if the dust collector is rigged to start simultaneously with the saw; fix that by starting the DC after the saw is spun up.
Other loads might be heaters; which can simply be turned off while the saw is running. Whatever power the saw is drawing, that turns into heat in the room. But only you know your wiring.
There also might be other loads inside your house you are unaware of. These lose power when the breaker trips.
Really, you are just putting way too much stuff on the average default typically 15A garage circuit. You need to run 2-3 additional 12 AWG circuits from the panel to the garage. (the hard part is working out the route; once that is done, running 1 or 4 circuits is the same effort, the only difference is cost of wire, and that's cheap). The saw needs its own dedicated circuit with a single (1-socket) plug. Why? Because it's a big load that really needs that. Also being on its own circuit will let us exploit an NEC rule if it keeps tripping breakers. You may be able to properly use that 30A breaker after all. Or you could convert that dedicated circuit to 240V if you can rejumper the motor to 240V or get a bigger saw.