I wouldn't subscribe to the "15 years ago" statement, but we have figured out mobile battery use pretty well, and the respective solutions are avialable in both server and client implementations. Have a look at [0] and [1] for (slightly biased but more detailed) elabortations.

simply declaring it a myth is, I suspect, unlikely to convince remaining skeptics

It's hard to provide specific counter-arguments to anecdotal evidence repeated since back when gtalk was a thing.

[0] https://gultsch.de/xmpp_2016.html

[1] https://wiki.xmpp.org/web/index.php?title=Myths#Myth_Three:_....