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Owning a domain name is still a reasonably good idea, but it was never a path to fame, even back then. A stepping stone, maybe, but even then, not strictly necessary.
Yahoo mail still exists - you can even set up a new one - but they got rid of the forwarding feature for free accounts a few years ago, so you actually have to make an effort to log in to the site and check. There’s no way to access free mail accounts other than doing that. Of course, if you’re willing to pay something, you can get forwarding back and maybe something like POP or even IMAP.
Of course, if you do pay, you’ll be supporting some of the more questionable decisions made by Yahoo management.
I think the point of doing this first is not about getting fame, but to be ready for it. Once you are famous someone can claim the domain you wanted and hold it hostage so by doing this you don’t have to deal with that later on.
There are many Android Apps that connect to yahoo through the web interface and retrieve your email so you can read it in a normal place.
I can access my mail on my yahoo address from fairemail on my phone and I’m not paying, isn’t it through IMAP? I don’t remember, the setup was very easy tho.