wrote

If it's the button, why does the light go off when pressed? Seems to me
it's
a high impedance somewhere, the transformer perhaps.

It goes off because it breaks the light contact, but does not make a good
enough contact to ring the bell. BTDT

My doorbell was broken for 20 years and I finally fixed it. Thankfully it
broke again; this time it will stay broken. People that I want to see know
to come to the side door.