The word "to" isn't just used before infinitives. It has lots of other uses as a preposition, generally before noun phrases. Since "ringing" and "watering" can be noun phrases, "to ringing" and "to watering" are correct in some circumstances.
For example:
This trouble can be almost entirely eliminated by careful attention to watering and ventilation. (Market Growers Journal - Volume 16 - Page 43)
Set ... to ...ing
Both of the example sentences you gave use the structure "set [direct object] to [verb]ing." In this construction, "to" is a preposition. There is no general reason it would not be able to be followed by an "ing" word.
I also don't know of any specific objection to this construction with the verb "set". An example is listed without comment in the American Heritage Dictionary's entry for set:
The noise set the dog to barking.
Since there is no reason I know of to object to this, I would say it is correct, and so are the two sentences you mentioned in your question.
I don't see anything wrong with "sets all bells ringing" or "makes my mouth water" either, but I wouldn't say your proposed changes are necessary.