It’s a rubber seal to help retain lube at end of pin.
Not arguing, just discussing.
If the bushing is there to retain lube at the end of the pin, why don't all the pins have the bushing, not just 1 on each caliper?
I was told by 2 different mechanics (one Honda dealer tech and one former Honda dealer tech who went indy) that the bushings are used for noise suppression. Again, why only 1 of the 2 pins on each caliper requires noise suppression is a mystery to me and to the 2 mechanics that I talked to. I'm sure there's some serious physics involved.
I replace the pins as needed and you can actually order just the little rubber seal if needed. My pins were fine but over time that rubber part swells and can bind worst case or break while trying to reinstall. Good idea to replace the pin seal and that small rubber seal after the second brake job. I can get about two sets of pads before they degrade.
Both mechanics said that when doing brakes on a customer's vehicle, they use/replace the bushings. Too much liability otherwise. However, they also both admitted that they have left them off of their own and family's/friend's car to no detrimental effect. As you said, if they swell and bind, they can prevent the pin from sliding. Once the pins stop sliding, they begin to rust. If they rust enough, you have the exact issue that the OP had: you can't get them out without a hammer. (Not to mention the pads that died early or the rotors that may have been destroyed.)
In addition, they both said that they have experienced the same thing that I have: The bushing sometimes hold air in the cylinder and will prevent the pin from staying fully in the cylinder. Push it in, it slides back out and stretches the boot, push it in, it slides back out and stretches the boot. No amount "boot bleeding" will let the air out because it's trapped in front of the bushing.
The first time that happened to me one of those mechanics (a neighbor) came over and said "Either try replacing the bushing or just take it off." Since no local store had the bushings in stock, I took them off and the pins slid in and stayed in. There was no noise and there was no rust the next time I did the brakes.
Disclaimer: I am
not recommending that anyone ignore all the research that the engineers did before they decided that some pins need bushings and others don't. That's up to each individual to decide on their own.