I think it's too early to tell based on one conversation the exact relationship between Death Watch and Children of the Watch, and their ideologies.. at least not enough to put in in this article as fact.
Personally I inferred from the episode that "Children of the Watch" implied that these are the successor to Death Watch, created after Death Watch's downfall (Hence, Children of the Watch, i.e. Children of Death Watch). My guess would be that when Death Watch fell, the survivors created the Children of the Watch who took on an even more zealous desire to return to the ancient traditions - and that's why the modern Watch doesn't remove their helmets. —Unsigned comment by [[User:{{{1}}}|{{{1}}}]] ([[User talk:{{{1}}}|talk]] • [[Special:Contributions/{{{1}}}|contribs]]).
Perhaps it should be added that in the new episode(s3 e1) the Children of the Watch are seen using the sigil of Clan Viszla. This combined with Din Djarin being rescued by the Death Watch, and ending up in the children of the watch, could reinforce the above theory, that they are the successors of the Death Watch.