To be clear: briefly pushing the button (and releasing) will cause the chimes to play their complete melody (?)
While not 100% conclusive, this suggests that the diode is intact and is allowing "half" the electricity to continue to flow even after the button is released.
You could *remove* the button and just try tapping the wires together (holding them by their insulation -- despite it's relatively low voltage) to see if the doorbell plays its entire tune -- or, if it just gets started and then quits (because there is no diode present with the button removed!)
Do you still have any of the old (toasted) buttons? Are they just "generic" buttons -- nothing ornate/fancy?
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