I see the Gaggia Classic Pro has some issue with the temperature as it uses a hysteresis to do bang-bang control. This is not necessarily something bad but from what people say the bands are quite big.
The common solution to the problem is to buy a PID kit with display, possibility to change the temperature setpoint and a shitload of crap.
However, to fix the problem you could just change the hysteresis limits so the bang-bang controller is faster to react and there’s less temperature difference. I have not looked into how to change these but I would imagine it would be a matter of some small soldering and a few cents of components vs 100 usd. Perhaps you can get away without soldering? Has anyone looked at this option? Of course you will use more the relay and may need to change sooner but these are also dirty cheap.
I’m not sure whether people have give you the answers you’re looking for, but a slightly different response: there are cases where people have installed very simple PID algorithms which dispense entirely with the ID part and only do the “proportional” part. These work well!
So you don’t need something complicated. However I still think these algorithms will instill a proportional response (so time of input will be proportional to the error) and this is crucial.