That's usually the first thing I try (because it's the cheapest fix) and then I'd start a bit more detailed debugging. I can't remember the specifics on this machine so it might not have a 3-way solenoid but if it does you could try backflushing with pullycaf to see if this makes any difference to the wand waterflow. If it does then it might be the 3-way clogged with scale or coffee.
But the cheaper Gaggias don't have a 3-way so check first.
"Gaggia Classic Pro: How to Clean 3 Way Solenoid Valve
If NO WATER comes out of the steam arm, and NO WATER comes through the group head, but you DO get water out of the decompression duct (the pipe on the front of the machine) then it is time to replace the solenoid valve."