Taylor was born in 1826, which means he spent at least a decade and a half living on post, on various frontiers and in DC, as his father climbed the ladder; and didn't Taylor serve as an ADC to his father, at least early on, in 1845-4?
It's not a formal military education or training, but it would certainly suggest some level of experience, if only by example. Most Army brats I've known could probably call cadence in their sleep, if you get what I mean, and would have a basic understanding of organization and administrative process, for example, much more so than their peers who never spent any time on an active post.
It's a slender reed, undoubtedly, but if the sieve is "no prior military experience" ( as opposed to military service) Taylor doesn't "really" qualify...
Best,