2000 Mazda Protege idling rough

I have a 2000 Mazda Protege ES (1.8L) with 120k+ miles and an automatic transmission. The car drives well at speed, but when sitting at stops it will shake a little, sometimes worse than others. Often the RPMs drop to below the lowest tick for a split second before recovering, very much like when driving a manual transmission and not pressing the clutch in enough. Even when the RPMs are steady at a stop, it shakes enough to rattle a couple of the plastic parts in the dashboard.

Recent history (some likely unrelated, but I’m no expert):
New air hose (one from the air filter to the engine) in September. RPMs were falling at idle more often than they are now, there was a decent sized crack in the hose. RPMs stopped falling as often, but has idled roughly before and after.
New radiator and thermostat in April (overheated on the interstate and cracked open).
New spark plugs and wires last fall after one cylinder wasn’t firing.
For the last couple years every fall when it starts to get cold, the alternator belt squeaks and I tighten the bolt a little each morning until it stops (usually after 2 days, maybe 3 - the alternator itself is on a long bolt that adjusts the belt tension).
Every 5k miles I have the oil changed with high mileage oil and tires rotated/balanced.

I took it to a garage a couple weekends ago after the check engine light had come on earlier in the week. I reset the computer and the light never came back on, so I lost the chance to have the code read, but there weren’t any other symptoms beyond what I’m used to. The mechanics at the shop said they couldn’t find anything specific causing the rough idle and suggested replacing some or all of the motor mounts, which are weak. They didn’t seem confident that would fix the problem, and neither am I, as I don’t see how weak motor mounts would explain the low RPMs as well as the shaking.

Any thoughts? Thanks for your help.

Idle Air Control Valve. They are not inexpensive so you might want to try and clean yours first.

“check engine light had come on”

Please tell us what the code was

Considering the overheating history and cracked radiator I might suggest a compression test to determine if a cylinder, or plural of that, has dropped and you’re facing a major engine mechanical fault.