Jet engines have on the order of 10x the specific impulse of a chemical rocket.

Atmospheric density reduces exponentially with altitude, which implies that you would need to go exponentially faster to maintain mass flow into your engines and lift over your wings. The truth is that breathing air only gets you a third of the way to space, at best, so you have to have a rocket, and now you're battling that complexity. If your space plane doesn't breathe air, it probably is just better to punch your way out the way conventional rockets do.

Of course, the rocket equation is logarithmic, so reducing the amount of mass you loft gives you an exponential gain. This is true for all propulsion systems to an extent (different constants) but getting into space is the hardest propulsion problem we face. A space plane may or may not be better in this regard (it's been a while since I've looked into that kind of thing, so no opinion either way) but imo the inherent complexity is enough on its own to kill the idea.