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.
So here’s a simple diagram of the Gaggia: https://imgur.com/a/445X7xq It seems it would be the element 4 - coffee thermostat - that does everything. A quick google search shows that you can find some for like 5 usd: https://www.bluestarcoffee.eu/en/Gaggia-Coffee-Thermostat-107C/m-435.aspx. Note that they sell them with different setpoints therefore the whole thing must the the control and there should be some available with narower hysteresis bands maybe.