Yes, that was more or less my point. And also Uber (more than Lyft) it's by far the major player, it's entrenched in people's imagination as the representative

Yes, that was more or less my point. And also Uber (more than Lyft) it's by far the major player, it's entrenched in people's imagination as the representative of the very service it provides, and it just works. "Call an Uber" is as idiomatic today as "Google something".