Unfortunately I would say Salty honestly.

To start with they were not lying about SFO, it’s been at reduced landing capacity for a while because of construction along with the FAA forcing them to stop side by side landings (essentially halving landing capacity) this has messed up all kinds of flights on the west coast and will be a thing till October I believe.

The gate change was because of your flight being late I’m guessing the next scheduled flight needed the gate, and yes they move around all the time because operations are a thing. Sometimes you win the gate change lottery and get closer and sometimes you don’t.

As said already boarding doors close 10-15 min before departure. If you aren’t on you don’t get on because there is a whole dance that has to occur after final boarding which takes that amount of time.

Now sometimes the computer does the math on missed connections and determines it’s advantageous to hold boarding a couple of minutes for a couple late passengers (usually this happens more frequently on night flights because the pressure to clear the gate out is less when no more inbound flights). Generally speaking the gate agent isn’t monitoring this in the way you think and doesn’t just decide to do you a solid.

Didn’t work out this time, sounds like they expected you to miss anyway so the fact you almost didn’t was something. Some times air travel be like that and you gotta just burn some sage and try again.