There are reasons for a lag time in the system. The real question is what are the actual temperatures during warm up, 30 deg spike....and final running temp. It would be nice, before you drill the hole, to test the thermostat on your stove with a good thermometer. This is a must do on a $6 Chinese thermostat. Which before box price, shipping, taxes, mark up and taxes again was 23 cents to make including material and labor.
The initial water pressure fights against the first opening of some thermostats. If the temps are indeed within safe range during this period, it's no problem.
The stove test will not replicate the opposing pressure but will indicate the static lag time and the calibration of the stat. The stat should break the seal at no more than 4 deg from rating. It will be fully open at 15-20 ABOVE rated.
Source: Stant
Drilling a hole is a band aid and may cause heat problems. There are a lot of things that do work OK. They were discovered by folks who lived far from town, low on cash and time. I'm not against what works. Lots of innovations were discovered that way. Lot's of folks took out the stat also, that's not good, but is still advised now and then.
