Well met brother,
I have just embarked on a Primaris crusade myself (with some Terminators sprinkled in mind!).
I think BT would take any advantage they can get, including Primaris. Doesn't make them fully paid-up members of the Codex club, they can employ them in their own way - getting up close and personal, but plenty of shooting on the way in.
In terms of battlefield usage the sky's the limit really, I've just gone with the models I like - provided you are unafraid of letting them knock some heads together in combat, pretty much anything goes!
When I first picked up my Templars again last year, I had been away from the game for a long time, so was initially resistant to incorporating Primaris. It took awhile for me to realize I was coming up with fluff justifications because I was letting my inner grumpy grognard show. Now, I'm more in line with your thinking. Having bolter-armed Primaris isn't even that big of a deal, fluff-wise. In the 4e codex, there's a little sidebar going on about how the Holy Bolter is the most perfect weapon, etc. etc. I do wonder how well bringing in the Unnumbered Sons would go (could be a fun bit of fanfic to write), but raising new Primaris or crossing the Rubicon should be no problem, imo.
While I largely agree in principle, I also would like to give a reminder that in the 4th edition Templar codex, Sword brethren veteran squads could be upgraded to infiltrate on the battlefield.
I think most things (except for witches, pfteeh!), even vanguard can be made to fit in a Templar context. Some things obviously fit better than others though.
Agreed here, as well. I've seen a few references to the Templars eschewing indirect fire weapons, but I can't recall what (if any) official source they come from. If someone knows offhand, I'd like to look at it. I do know we didn't get Whirlwinds back in the day, so maybe it's in the 4e book somewhere. As far as the Vanguard marines go, I got some because I like most of the models (looking at you, Suppressors), but was torn about integrating them because sneaking about felt un-Templar to me. I still may do them up as Celestial Lions, but I am backing off of that assessment.
For me, it all goes back to a couple of things. The Marshals of the crusades, zealous fellas that they are, are still Astartes commanders with centuries of battle experience, so are not going to eschew a superior tactic that complements their style. Plus, especially in the case of Incursors, that sneaking about is actually just a really good means of getting to grips with the enemy and opening breaches in the line so your chainsword-wielding brothers can pour in. In game terms, being able to get a turn 1 charge off fairly reliably with a unit that has a decent combat bonus for chewing up chaff units seems pretty good. Eliminators are a tougher sell, because Templars with sniper rifles still feels a little weird to me, but maybe I'll come around on that as well.