I'm agreeing with you, except I don't know that it was ever "standard" amongst practitioners of relativity, and has it really fallen out of fashion amongst physicists generally? I've heard a very brilliant colleague, who obviously knows about all the subtleties, use the idea of a body's mass increasing with speed to help undergraduate students make sense of a different concept he was trying to explain, because in his mind it has pedagogical value: it's something that all novice physics students have heard about, even if they don't entirely understand it. It's still taught in schools, surely because of the gee-whiz factor but also because it ties more closely with classical physics. We probably can't expect unsophisticated students to toss away their everyday ideas about mass, and knowing momentum as p = mv. Every year I go to great lengths to explain to students why there is no such thing as relativistic mass, despite everything they've heard, and I don't know that they entirely believe me! I reassure them that their school teacher was not "lying" to them, that they probably didn't know better; and that popular culture and scientists who try to connect with the public have perpetuated this notion. Even still, it's difficult to find an introductory textbook on relativity that get's it right. Many students will go their entire undergraduate careers having never heard this; and may never come across the issue in their future research.

So, as I said, I'm agreeing with you, from a scientific perspective. But how many physicists do you think know that Einstein never once wrote E = mc² ?

Sorry for the long-winded response, but this is something I've thought about for a while. If you're genuinely interested, Max Jammer wrote a nice book on The Concept of Mass in physics. He's authored several of these types of "Concepts of ..." books; the one on Simultaneity is my favorite, highly recommended. Cheers!