It seems likely to me that having the Armorer recap the whole history of the Darksaber was to establish specifically within the narrative of "The Mandalorian" that someone can faithfully be both a Mandalorian and a Jedi.
I guess it depends on how one defines those terms. Mandalorian can just mean a person born either on Mandalore or one of it's colonies (or decended from such people that consider themselves culturally Mandalorian), or it can mean anyone initiated into the cult of The Mandalore and follows the creed. The latter of course believe that they're the only ones that count as Mandalorians, regardless.
If we're just talking about the former then yes, they can of course be inducted as a Jedi just like anyone from literally any other planet can be. Also, the Jedi appear to encourage their Initiates to both understand and embrace their unique cultures of origin (since they're taken so young, it's not like they'd remember or have any clue otherwise.) So in that sense there can of course be Mandalorian Jedi in the same way there can be Alderaanian Jedi, Kashyyykian Jedi, Coruscanti Jedi, Rylothian Jedi, Mirialan Jedi . . . etc. etc.
However it seems like the creed of The Mandalore and the Jedi Code are mutually exclusive, if not diametrically opposed. So embracing one should necessitate abandoning the other by sheer definition.
We still don't have the full story of Tarre Vizsla yet. All we know is that he was born Mandalorian, became a Jedi (building the Darksaber in the process), wound up ruling Mandalore with the Darksaber in hand; apparently founding House Vizsla in the process (sorry
Shae!), died and either bequeathed the Darksaber back to the Order, or they recovered it by other means, only for his decedents/inheritors to break in and nick it in the chaos of the Fall of the Old Republic and the Rise of Darth Bane's Empire.
We don't know if he (somehow?) remained a Jedi while ruling Mandalore, but it seems unlikely. Even as a Wayseeker there would have to be limits that the High Council would tolerate and "ruling a planet of conquering warriors known for going to war with Jedi" seems just a
teeny bit over the line, no?
It seems more likely he chose to leave the Order out of some sense of obligation (last of his line, the only one that could unite the houses and end yet another Mando civil war, that kind of thing.) And out of respect, the Council permitted him to keep his sabre (as they did with Dooku and Ty Yorrick.)
Which also leads to the question; what is a Jedi? Usually that's pretty cut and dried; a Jedi is anyone officially Knighted into the Order. However post Order-66, it's a but more flexible. Ahsoka for example left the Order, but her actions, how she comports herself, and the code she holds herself to makes it a valid argument that she is a Jedi in all but name. Perhaps even more so than those that were actually Knighted. Even Luke was never officially Knighted since there was literally no one left to do it. So does the philosophy matter more than the formalities? You would think, but it depends on the when and where of it all. Context matters.
This is a long way of saying that I don't think the trajectory of this show is "how Jedi Initiate Grogu became a Mandalorian of the Creed", but more likely "How Din Djarin redefined what being a Mandalorian means to himself."
In that sense, he's on a similar journey as Sabine, who it can also be said is reconciling Jedi teachings with her (much less fanatical) Mandalorian heritage.