My house was built in 1929. The doorbell is presumed to have come later, but maybe not much later. It had been getting a little unreliable the last few years (!!!) but it all came to a head two weeks ago.
Anyway, the first step to healing is to admit you need to disassemble the thing.
And a detail of those switch contacts:
So, those machine screw ends are shorted together when the doorbell button pushes the brass spring down onto the (corroded) ends of the screws. Repair methodology: file the metal parts until they’re shiny and new looking. The material that the threads screw into is about 50 laminations of plain old paper. Amazingly, it has held up through at least sixty years of outdoor use.