Yeah, it gets worse than that, too, because some airlines have multiple
types of gate checks. Delta, which I've flown the most on recently, lets you gate check at the gate desk up until like 10 minutes before boarding and those get tagged like normal checked baggage, sent to a baggage car for the normal forklift-ability, and will show up at baggage claim at your final destination, but Delta
also lets you gate check at the last minute by dropping it at the end of the jet bridge and those get a slightly different tag, are basically thrown into cargo by hand, and also need to be pulled out of cargo by hand and so are thrown right back onto the jet bridge on the deplaning side of the next destination, not the final one. In the case of "overhead bins full" every bag after that full point gets into that second category of jet bridge bags. (And yeah, I've seen that particular case cause some traffic jams in deplaning.)
(Delta also has a third type of "gate check" if you count the regular checkin desk silliness to try to skip bag fees by checking in with 0 bags, getting asked if you want to gate check, and then checking it at the checkin desk like checked baggage is supposed to work. That also goes to your final destination, but it's a silly process of "no I don't want to check bags" to say that "yes, I have one bag I would like to check but it's not worth your silly fees to check if you want to charge me".)
I also have met people that like the "jet bridge checked baggage" and think it a feature, not a bug. I understand there is a flexibility it offers if a connection fails or is too delayed or what have you, but the slow, artisanally hand tossed baggage part of that seems so inefficient and slow down to the rest of us, it is hard for me to not see that as a bit selfish and something that should have fees to pay for the extra labor and time involved. Also, if anything it seems a reminder that Baggage Claim got put on the wrong side of Security checkpoints in the US out of a mistake from historic airport layouts, and if you were to design the system from scratch you'd put it before leaving Security and allow people the option to choose which destinations it needs to be picked up (but still defaulting to the final one), and maybe a "recheck" desk right next to it.