Why install a full blown PID in Gaggia Classic instead of changing the hysteresis limits?

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.