I have an old fashion ding-dong doorbell that is powered by a 12VAC 1A transformer. The old transformer died and I decided to replace with a cheap AC LED driver which is rated 12VAC 50W. That didn't work. Then I bought a classic bulky 12VAC 1A transformer (similar to what I had before) and that worked.
Why did not the LED driver work? It's AC (not DC, verified with a multimeter) and it works well when I test it with a 12V MR16 LED down-light bulb.
Probably your doorbell requires AC at 60 Hz frequency; to save weight and cost, an LED supply might use a high frequency (50 kHz) and very small transformer to power LEDs, but this will more likely demagnetize the doorbell's moving part than it will drive it.
A classic doorbell magnetizes the iron part with electric current, taking about 1 millisecond. Then it reverses the polarity a few milliseconds later. But, it's usually completely magnetized while the AC is switched on. That same doorbell, with 50 kHz excitation, will only achieve 1/100 of its full magnetization before the electricity reverses and tries to magnetize it in the other direction...
The transformer you see in the doorbell is an Step down transformer designed for power supply frequency
where as the transformers used in LED drivers are SMPS (switch mode power supply) transformers ( they work at very high frequency) --- that is the reason you can observe that they look very small when compared to a regular transformer.
