EV and ICE have great synergy if you use them together. EV offers amazing acceleration from zero, and relatively less amazing acceleration at highway speed. ICE has better acceleration when you're already moving. ICE efficiency is poor at low speeds, and better at high speeds, and EV is opposite. ICE can warm up the interior of the vehicle at no extra charge. If the engine is idling to warm up (itself or the occupants), might as well charge the batteries. If the battery has enough charge, it can easily start the engine when needed, so you don't need to idle unless warming is needed... And EV is always ready so you don't have the delay to go that some auto start/stop systems have.
Anyway, at least in my PHEV, it had a 'EV only' mode you can use if you charge it, that operates close to what you suggest. It does run the engine a bit here and there for 'normal operations' and will try to cycle through the gas tank over several months to make sure the gas doesn't go stale. But if it's in EV only mode, acceleration from say 40 mph to 65 mph is poor, and the battery drains quick at 65 mph vs 30 mph ish or stop and go. Significant uphill eats the battery too.
The one I drive is also very much a compromise vehicle, the extra battery for PHEV doesn't have a good place to go, so the trunk space is poor. There's probably ways to design a car to be PHEV first and find better places for the battery, but most/all? PHEVs I've seen are designed hybrid or ICE only first and then stuff extra batteries in storage areas.
Many good options abound, but they’re all $10-15k more than the basic guzzler. And $15k is a lot of fuel.
This is the default setup for mine, it starts on battery unless it's too cold (e.g. today's morning was -22C). Li-Ion batteries don't like cold (or heat), esp. don't like charging at negative temperatures.
In short they are a thing, and a rather pleasant one; as a side bonus the combined power of the two engines is quite nice, e.g. taking large trucks at two lane roads (pedal to the metal does engage both engines)
YMMV if you have a super long commute.