Combustion Man background was in various official media. The wiki sources the old ATLA website on Nick.com, which I remember, and I think it's been in interviews too. Where did you hear about a military experiment?
I already gave an explanation for why they'd have the tattoo: hubris and tradition. It would also command respect and foster fear in anybody who saw it, which can create a psychological advantage before the battle begins. And the wiki doesn't actually say that the tattoo is necessary -- just that the tattoo is there.
Re: the tattoo as an aid, I did ask if this was what you meant way back at the start.
I don't think it makes sense to assume that the tattoo confers power or focus that the user didn't have innately or through training. There's no logical basis that a tattoo could help focus a beam. It's not a lens! At best, I could buy that it's a psychological thing, like when (in Naruto) Jiraiya drew a spiral on Naruto's palm to help him learn to do the Rasengan.
I'll point to the only other case of meaningful inkwork in the Avatar universe -- airbending master tattoos. Those tattoos are given because of the user's proven ability. It makes more sense to me that the combustion tattoo is the same. To me, it's much less of a stretch or intuitive leap that their power is innate, since there is precedence for that too in all the special forms of bending. Those are abilities that are rare among benders.
But yeah, we can agree to disagree. Going back to the original statement, the source of combustion
isn't explained in the series, much like lavabending. And I think that's alright.