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.
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.