Ive been using Memmy and Mlem and both are great. I think overall Memmy feels a bit more ahead in terms of features particularly after the latest few updates. The main think I miss in Mlem is the availability to zoom in on images and in Memmy a way to change font size.

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.