Sometimes that analog persistence, or its equivalence in DSOs is useful and that is probably why more expensive modern scopes try to emulate that. Not showing switching noise may be wanted effect, if it is known to be external, non-removable and unimportant factor. And there is also difference in analog scopes, my scope shows often strongly that noise(cheap UPS is huge noise generator), and sometimes I have mistaken that noise to be coming from actual device under test, which gives gray hairs.
Thanks Dave for making these vblogs, now I know to start looking elsewhere when I see that next time!