In a nutshell, the New Mandalorian movement began after "The Mandalorian Excision" -- a lovely clinical government euphemism for when the Republic pre-emptively bombed Mandalore into the desert we see a few centuries before the movies. The rest of the damage happened during the Mandalorian Civil Wars. There was a lot of internal dispute about whether the Mandalorians should be constantly-warring conquerors, or noble warriors who hewed to a code of conduct. Eventually those factions killed enough of each other the New Mandalorians came into ascendency, banned the wearing of the armor, and adopted an official policy of pacifism and neutrality. They'd been getting dragged into galactic conflicts for three thousand years and it was destroying them.
That decision was not embraced by everyone. The so-called True Mandalorians refused to give up the armor and maintained the Mandalorian Code. Over time, they got squeezed out -- exiled to Mandalore's trojan-orbital partner, Concordia (much smaller, but much denser, due to its beskar deposits)... or finding refuge in outer Mandalorian worlds like Concord Dawn. A reactionary group formed as it seemed the old ways were on the way out. They called themselves the Deathwatch, as an expression of how they felt they were seeing the end of what it meant to be Mandalorian.
After Deathwatch was co-opted by Darth Maul during the Clone Wars, many members left (among them Bo-Katan and Ursa Wren), and eventually the rest of Deathwatch were wiped out. There is a piece of the narrative still missing, so we can only speculate, but some portion of Deathwatch survived, neither rejoining Mandalorian society nor staying with Maul and being imprisoned or killed. They were hardliners. Nationalists. They cranked all their Mando-ness to 11. They had kids and passed their tenets on to them. Din was a Foundling they took in during the Clone Wars. They presumably already had kids, and likely proceeded to have more in exile. We don't know if Din's covert was the only one. We don't know if the Purge was something imposed on the Mandalorians by the Empire, or if it was something the Mandalorians did to the remnants of Deathwatch. We just know they were in a self-inflicted information blackout and felt, for whatever reason, they could never go home. Over the quarter-century since they left wherever, they have adopted this extremist interpretation of things as pretty much a religion.
Other Mandalorians do and have always taken their helmets off as needed/desired.
