Gmail.com is certainly broken in the sense that they want to cram 10 billion users into a single domain. It's ridiculous marketing/brand-motivated UX failure.
Since forever most mail services had a few vanity-domains, so people could get first.last@wherever.com. But no, Google doesn't want to provide email, they want to provide "Google Mail".
Apologies for the rant, but I can't stand it when big companies create problems through stupidity.