Some hybrid cars almost work this way. I know at least Honda's hybrids basically do what you're suggesting but at constant highway speeds will directly couple the engine to the drive wheels. They presumably could use electric motors powered by the engine in all driving scenarios, but I believe direct engine drive at highway speeds is more energy efficient.
This is probably why most hybrid systems I'm aware of don't only use electric motors to power the drive wheels. The idea sounds cool and I've also wondered why you can't buy something like that in the US (I think it exists elsewhere), but the math doesn't really work out. Even in terms of engineering complexity, because the engine is only directly driving the wheels at certain speeds, you can get by without a lot of the mechanical drivetrain components like transmissions.
They exist, but I don't think there's currently any new models for sale in the US. I think they're generally called "series hybrid," or sometimes it's marketed as an electric vehicle with range extender.
I believe the Chevy Volt worked this way - you can see used ones for sale for around $15k.
