On the waterproof-ness of Banglejs 1

I did not rinse it with clean water, do you think that could help? If so, how?

I would definitely give that a go. What can happen is even when dry, stuff gets deposited on the PCB or between the connections on the chip and that can be conductive enough that it connects the pins together and causes problems. Or worse, if it's wet and there is electricity the wires can get electrolytic corrosion and then there really isn't much you can do.

After giving the PCB a rinse (try not to get the battery wet if you can), leave it to dry somewhere warm (and try as hard as you can to make sure it is dry) and maybe it'll come back to life again properly.

I found very hard to put back the rubber waterproofing ring in the correct position. It seems to me that its diameter is a couple of millimeters short of what the bangle would need

It is possible it has contracted slightly over time. However when closed it should have made a good seal. I have seen Bangles where the rubber ring wasn't in place, but in that case you should have noticed that the rubber had been crushed in some places.