I've taught a treatment of series that was similar to what you are describing, if not quite as abbreviated. The primary difficulty I would point out is that you need the root test for  a "good" (constructive) proof that Taylor series have a radius of convergence. There's a correct but completely non-constructive proof using I think the comparison test. If you actually want to compute the radius of convergence in real examples you of course use the ratio test; the problem with using this for a proof is that it fails if the series has terms equal to zero. 

I'd also point out that if you want to prove that the ratio test works, you need some of the other tests. (E.g., if I recall correctly, this can be proved by showing that the geometric series converges for ratio less than one, and then using the comparison test.)

Nevertheless, if you're willing to handwave a lot of stuff, something like this can be made to work. My only other comment is that you should be wary of trying to rush through Taylor and Maclaurin series: a lot of students have trouble with these, and they are quite important in subjects where mathematics might be applied--much more so, I believe, than convergence tests, polar coordinates, or arc length.