You're not totally right. A jet engine can provide a large amount of the ∆V required to get into orbit, so your orbital injector could be significantly smaller than a full orbital rocket.

Yeah, a mixed-mode engine is probably not the ideal. You'd probably just use a liquid fuel rocket carrying oxidizer, like every other orbital rocket, as the payload on the jet, but you'd get to skip a large portion of the launch. Keep in mind that the rocket equation means that most of the fuel is used for the initial portion of the launch. If we can have an air breathing jet handle that then the rocket can be significantly smaller.

If you're just trying to insert a single small payload into orbit, it makes some sense. It's very limited by the capabilities of the aircraft though, obviously. A rocket can do significantly larger payloads with more versatility, so it ends up being more practical per ton of payload.