We do have a pretty extensive greyhound network in the US (it was recently bought by flixbus, but I think they're keeping the greyhound branding). But it is a totally different experience than long-distance buses in Europe.
I've taken a number of flixbuses in Europe and they were decent. Most cities had some sort of bus terminal where you could sit and grab food before the trip, the bus was clean, the advertised power outlets worked, seat assignments were respected, and they'd make stops at places with clean restrooms.
I've also taken a few flixbuses in the US west and wouldn't recommend people do that. The cleanliness is poor, the on board amenities like power outlets/wifi are frequently broken (or don't exist), paid seat assignments have never been enforced, bathroom stops are disgusting, and there is a lack of order/safety on board. And the bus terminals simply don't exist - in Los Angeles you stand in an uncovered parking lot outside the central jail and there is no notification of which bus goes where. Everyone just runs (literally) up to each bus as it arrives, preventing passengers from disembarking, and asks the bus driver where it's going, and then tries to cram on immediately because that's the only way to avoid getting an undesirable seat.
Safety is a huge issue on these buses. I conduct most of my life on foot and by public transit in southern California and there's some sense of safety from knowing you could avoid/move away from people acting dangerously. But on the bus you're stuck, and I often arrive with my nerves frayed by the behavior of other passengers. They don't publish stats on this but my friends who have taken these buses also reported they also feel extremely unsafe, with one telling me they were on a trip where a stabbing occurred.
I regularly take the Amtrak from San Diego to LA to see friends and family, and during the track closure last year I tried flixbus a few times. After 4 tries, I simply stopped discretionary travel to LA for the better part of a year because it was such a bad experience.