The Armor priestess mentions that having the full beskar will stand out.
Yes, but she may simply mean that it will stand out in comparison with his previous panoply of (duraplast?) armor (which would make sense, since he's such well-known figure, apparently).
This seems to imply they do not paint it.
Not necessarily. As I've noted, the show cribs a lot from Karen Traviss for its depiction of Mandalorian culture, and one of the more "colorful" elements of her take on Mandos was that they almost always paint their armor in very distinctive ways, each hue communicating a different message about its wearer's mindset, intentions or personality. Blue, for example, was supposed to indicate a Mandalorian who cherished "reliability," and that does seem to fit the Heavy, who comes to blows with Mando over a point of cultural decorum but is still willing to charge in, giant gun blazing, to haul his ass out of the fire when it counts.
I doubt he would purposely leave it so, its out of his character to want to stand out.
It could potentially tie in to his state of undecidedness that I commented on. He's not comfortable accepting a Mudhorn as his totemic "signet" because it "wasn't a noble kill." By the same token, he may not be comfortable painting his new armor because he doesn't have a signet yet (although the out-of-universe reason is probably because the silvery
Beskar makes him look more like the proverbial "Knight in Shining Armor").
They also go out of the way to mention that he is a foundling. Which could lend credence that he isnt actually mandalorian by birth(not like thats a super huge deal for mandalorian culture it seems)...
Well, he outright states that he's not Mandalorian by birth in "Sanctuary" (which potentially adds an interesting additional layer to the scene in "The Child" where he determinedly gets back on the Blurrg after Kuuil invokes his Mandalorian "ancestors" taming the ancient Mythosaurs)
It would be interesting to get some more stuff on the purge. Since they imply the empire did it, but his homeworld was ruined by separatists.
Well, again, his homeworld was apparently not a Mandalorian one, but yeah, they seem to be drawing from Karen Traviss again, and her telling (ultimately inspired by the old Marvel comics) of Mandalore being occupied by the Empire after the Clone Wars and strip-mined of its
Beskar.