There are a lot of different versions of the Gaggia Classic, but the versions I find most interesting are those sold in Europe. As far as I'm aware (and I've actually never owned or used a Gaggia machine) the European versions all include a standby timer, essentially a device that will automatically switch the espresso machine off after a set amount of time. A steam boiler is an inherently risky piece of equipment - in the very best case leaving it switched on for a long period will waste a lot of electricity, but the worst case could vary from burning out the heating element, to causing a fire, or even causing a steam explosion. Unlikely, yes, but conceivable. There are safety features to prevent this like a thermal fuse and pressure release valves (I don't know for sure if any particular espresso machines have PRVs), but I can't see any reason to forgo the additional safety feature of automatically turning off... before anything like that can go wrong.

The Breville 800ES/B had this feature built into it's electronic circuitry, but this was all removed, so it's a feature I had to add back in. To do so I used this time-delay relay: https://www.aliexpress.com/item/32881831418.html?spm=a2g0o.order_list.order_list_main.94.54861802tmytIp. Among it's many functions it can be configured to close the relay when it first receives power, then automatically open the relay after a configurable amount of time between 1 second and 10 days. Correctly wired this provides the automatic shutoff feature perfectly.

In this picture you can see the relay inside the coffee machine. It's mounted to a 3D-printed frame that also mounts the solid-state relay and a WAGO-style terminal block to distribute power. This in-turn is mounted to the screw bosses that were used to mount the original control PCB to the machine's chassis