Measure the voltage across the wires that go to the chime with one of the wires disconnected from the chime and when someone pushes the button.
It is possible for a loose connection or a chime that draws lots of power to show a voltage of 10V when everything is connected and the system really is a 16V system.
When cleaning anything including baby diapers, it is better to sweep off or scrape off as much solid and semisolid material as you can first. (1) Less cleaning fluid or detergent needs to be used, (2) Less of a mushy mess which in turn means that bleaches or disinfectants will be more effective.
In the olde days, many of the cheaper "builder's model" of door chimes utilized 10 Volt transformers. Installing a new 16 volt model on a 10 volt system will result in poor performance -- a weak sounding chime, if at all.
You SHOULD have access to that transformer. Do you know where it is, and why do you not have access?
The builder's models these days all come with 16 Volt units. I don't know if there are any of the older 10 Volt units available anymore ...