From memory:
Archangel Michael gave Sanya Esperacchius. Sanya has also referred to receiving directions from Michael multiple times.
Archangel Raphael is described by Mab as the "Demonbinder," and that he prefers to work with his own hands. The wards on Michael's house were, as I recall, described as Raphael's handiwork by Lasciel and Amoraccius has only gone out twice we know of- to perfect candidates each time. Possibly when the other two Swords were already in use? Regardless, Raphael rarely appointing a Knight tracks with preferring to do things himself
Archangel Gabriel is the Trumpeter, and Mab notes he's proclamatory. *Something* seized Murphy when she was wielding Fidellacius to pronounce judgement on the Red Court.
Mab notes that the Watchman, Uriel, is the most subtle and his works are rarely observed.
That's consistent with him not appointing a Knight.
I originally concocted this theory after Warriors, thinking that giving Harry Soulfire was anointing him as a Knight of the Cross, and was the equivalent of giving him a Sword. I still rather think this- it explains why Mab would "share" Harry with Uriel at all, if Uriel had a prior claim (even if Harry is ignorant of said claim). Once the Lance of Longinus appears, the idea of it being the fourth nail for the fourth Archangel simply fit.
I quite agree Harry is effectively an extra Knight - just look at what he does to Hannah Ascher in the vault, standard Knight working with added explosions rather than swords. Of course since Uriel is skirting the rules to get a 4th Knight, Lasciel still gets to be slightly manipulative even when Harry is talking to her, but that's the kind of tit-for-tat you get.
However, being that the case, it means Harry has been a 4th Knight since SF. So why would he need a 4th Sword to make him such? Nevermind that he has none of the three standard divine elements (Faith, Hope, Love) to work from.
However - there was a great big theological debate about Faith alone being what saves souls or Faith
and works doing that, so making Uriel/Harry the Knight of 'getting the job done according to faith/hope/love' does have a certain appeal.
Your theory also does have a bit of fridge brilliance to never clear it up, so we might never get a clear yes or no; according to Wikipedia, there's debate about whether the crucifixion used 3 or 4 nails... And of course Uriel being a 4th Archangel is also disputed outside Dresden lore (and there's 3-4 more Archangels if you want to get deeper into the lore, depending if you follow Talmudic, Catholic, Orthodox, Coptic, or Anglican ideas - but at least thanks to Mab's summary we can exclude those as particularly relevant; maybe they have jobs like keeping the stars burning rather than anything with humans).