Doorbell Voltage Drops at Transformer
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.