80C is really low for pulling shots. Normally it's 90C:
https://bigcupofcoffee.com/espresso-brewing-water-temperatur...
https://bigcupofcoffee.com/espresso-brewing-water-temperatur...
Also, a Gaggia Classic (as an example - they don’t specify which model they use) does not have a PID. Temperature of the espresso leaving the group head drops during the extraction.