I don't know what's with the Van Halen brothers, they seem to resent that Anthony has befriended Hagar more than anybody else before in that band.
Interestingly, this is what Roth has to say about Anthony:
“Clearly, vocals are every bit as much a component of success as a rhythm section or a guitar solo, and there’s an old expression saying, “They don’t go home singing the lighting show, they don’t go home singing the production.” You’re right, they sing my words and my melodies. And what we have at our fingertips is arguably one of the greatest high tenor voices ever – that was in Michael Anthony. In our tiny little corner of the universe, that voice is as identifiable as the high voice in Earth, Wind & Fire, as identifiable as the high voice in the Beach Boys. Van Halen is an indelicate house blend of both – that’s intentionally. So I would always look forward to that reunion.”Which probably hasn't raised his popularity stakes with the other two!
I think the comparison to EW&F and the Beach Boys is very apt - you might add Barry Gibb's backing vocals in the Bee Gees, Brian May's backing vocals in Queen, Timothy B. Schmidt's backing vocals in The Eagles and Phil Collins' backing vocals in early Genesis (imagine Carpet Crawlers without them, you can't!) -,
Anthony's voice was that much a trademark too.
When I heard the VH debut in 1978 at a friend's house, I noticed two things as "sounding different": (i) the guitar, (ii) the high-pitched backing vocals. Roth's verse vocals struck me as someone talking mostly over the music because he couldn't really sing (that's not a knock, he has qualities of his own, and the "talking" is very often effective - at least in the studio!), with the catchy chorusses making up for that (a recipe VH work to this day) and he looked, well, like Jim Dandy of Black Oak Arkansas! Van Halen really didn't have proper singing in their verses until the advent of Herr Hagar IMHO.
As for Eddie's vocals being part of that chorus sound - yes, of course, it was obviously always a very processed mix of voices (as it was with Queen or Abba too) and now that Anthony's tenor voice is gone the backing vocals still sound Van Halen'ish on the recent live album and that is very likely Eddie's voice, but there is an important ingredient missing, it is only an approximation.
As for the "teaching him to play bass"-comment: I don't doubt for a minute that Eddie has shown/taught MLA bass patterns or made proposals in VH's lengthy recording history - guitarists do that. And that is perfectly ok, I sometimes show them riffs and chords too, they are not always very good at those.
But on the same token I believe that MLA could come up with enough things himself. There are enough mannerisms in his playing on the VH and Chickenfoot albums to tell me that he a style of his own and didn't need videos from Eddie for that. The younger Van Halen brother obviously has a very selective memory. And a mean/bitter streak.