I feel like "to google something" has by now become a generic term that means just to search on the web, not necessarily using Google. In particular, in Russian, I've heard people say "загуглить в яндексе", literally "to google in Yandex".
By the same token, the users of XXX dot com will probably still refer to their posts as tweets. Though as OP points out, "tweet" isn't genericized like kleenex or google, so it is less likely to survive the sudden brand shift.
No one quite understands that. But if you want just the search, there's ya.ru.
However, in Russia, Yandex is not just a search engine. It does everything and then some — search, cloud storage, email, maps & navigation, music streaming, a voice assistant with Amazon-Echo-like speakers, news, online advertising, event tickets, a marketplace, taxi, food delivery a-la Uber Eats, car sharing, online grocery store, e-scooter rentals in major cities, and this list goes on and on. And that Dzen ad-filled cringy blogging thing no one seems to like. But Yandex is literally inescapable, it's everywhere and you'd be somewhat excluding yourself from the society if you refuse to use any of their services.
