Apparently, the new doorbell is made to operate at 16 volts not 10. It seems to not work well under-voltaged.
I would install the new transformer and disconnect the wireless extender for a while until you determine if the new door bell is working properly. Just make sure you shut off the circuit breaker feeding the old transformer before replacing it. Double check to make sure power is shut off before replacing it.
Good luck!
Assuming you have a full mains voltage going into the transformer, then there is a short-circuit in the wiring (or, much less likely, the transformer is vastly underrated for the current drawn).
"Bell wire" has thin insulation, compared to mains wiring. A staple may be pinching the wires, or wires could have rubbed together, tearing the insulation. Sometimes mice or squirrels can cause this. Such a short could have burned out the old transformer.
Another possibility is a short in the chime, which then caused the contacts of the doorbell switch to fuse together.
Also, inspect your connections to the transformer. If there are multiple wires, e.g. from both a front and a back door switch, they could be misconnected.
If you can, check the resistance between the bell wires going to the transformer secondary. The resistance should be greater than 16 divided by the current rating of the transformer, or (16 x 16) divided by the VA rating. For example, if the transformer is rated 10 VA, then circuit resistance should be greater than 256/10, or ~26 ohms. (Not to worry if it's just a bit lower, though: you're measuring DC resistance, while AC reactance across a coil is higher.)
Don't be concerned about the 22 VAC open circuit output because the nominal voltage, 16 VAC, is at the rated load.
BTW, my compliments on checking the voltage on installation! Had you not done so, the transformer would have burned out again, or there is a small possibility of fire if a "short-circuit" is a few ohms and is by something flammable.
Best Answer
The problem could be the button, the chime, the transformer (tested OK), or the wiring in the walls. Check the devices before chasing wires through walls. Start with the front button: disconnect and touch the 2 wires together. Does it ring? Then put your voltmeter on the chime coil. Does the chime get voltage when you touch the button wires together? If you have voltage at the chime with no bing-bong, then the chime is faulty. The only time you should measure voltage at the chime is when the front button is pushed. If it rings only when you touch the wires, then the front button is faulty.