I think the Chevy Volt is what you're talking about. You plug it in to charge and when the battery runs out, the engine runs to charge the battery so you can drive until the gasoline runs out. The engine isn't connected to the wheels at all. It only charges the battery as you drive.
I didn't realize it was that separated. Would be cool if you could detach the engine entirely and leave it at home when you don't need it. Or buy the car without an engine and rent one when you want to do the annual roadtrip
AIUI the Hybrid Synergy Drive combines aspects of a series and parallel hybrid, for efficiency at both high and low speeds. I think this is the main reason that pure series hybrids are not more popular.
See the Toyota Prius. This is already what is done.