I think you're greatly overestimating just how efficient a conventional ICE can be if designed and tuned for single-rpm, maximum-efficiency operation.
Basically, the Chevy Volt is exactly the car you're describing. It wasn't that great: it had a separate engine that only ran a generator, and would only come on when the battery was depleted. The problem is having a full EV powertrain, plus an ICE driving a generator, adds a lot of hardware, complexity, and cost to the vehicle, and the result was a car that people liked, but just wasn't cost-effective compared to the Prius, or to a BEV.
The Prius can get away with a much smaller battery because it's just a parallel hybrid and doesn't need to drive 50 miles on a charge like the Volt. Batteries are expensive and take up a lot of space. And a BEV can get away with not wasting space and weight on an ICE engine (plus fuel tank), and instead just have a big battery. The Volt basically had the worst of both worlds. Reportedly, the car worked well, but with all that stuff packed inside, it just cost too much, plus it didn't have a lot of cargo space.
Just FYI, the latest Prius Prime is functionally identical to the Chevy Volt. It just costs a lot less (starts at $33k). GM simply failed to design it properly.
Basically, the Chevy Volt is exactly the car you're describing. It wasn't that great: it had a separate engine that only ran a generator, and would only come on when the battery was depleted. The problem is having a full EV powertrain, plus an ICE driving a generator, adds a lot of hardware, complexity, and cost to the vehicle, and the result was a car that people liked, but just wasn't cost-effective compared to the Prius, or to a BEV.
The Prius can get away with a much smaller battery because it's just a parallel hybrid and doesn't need to drive 50 miles on a charge like the Volt. Batteries are expensive and take up a lot of space. And a BEV can get away with not wasting space and weight on an ICE engine (plus fuel tank), and instead just have a big battery. The Volt basically had the worst of both worlds. Reportedly, the car worked well, but with all that stuff packed inside, it just cost too much, plus it didn't have a lot of cargo space.