I am one of the ones using a cleaner with a tea tree oil in it to clean my cages. Firstly, it only contains a small amount of tea tree oil in it, and I am further diluting it before using it in my cages. Secondly, I've always followed through with a rinse (which is just common sense) with any product I've ever used.
I know a reputable breeder whose been using this same product around her chins for six years with no problems.
I think you can take this kind of thing to the Nth degree, being afraid to use anything around your animals.
The negative research that I saw was about using pure tea tree oil on your pets for wounds, or diluted in a shampoo, which would be directly applied to the animals. And of course that is not the case here.
Of course, like anything you do for your pets, this is all personal choice, and everyone should do what they are comfortable with. I'm just adding my two cents.
Whenever you change any aspect of your care, you should always do the appropriate research beforehand.