In sci.space.policy Jeff Findley wrote:

Who cares? What we're trying to optimize is cost, not dry mass of the
launch vehicle or any sort of efficiency you can measure on the launch
vehicle.

As Henry said, fuel and oxidizer are cheap, at least they are if you pick
reasonable propellants. A favorite of mine is LOX/kerosene. LOX is one of
....
Also, tankage is cheap and the thrust to weight ratio of a rocket engine is
better than that of an airbreather built to operate over a wide range of
speeds (subsonic to several Mach numbers). That and a rocket engine isn't
as complex as such an airbreather. Plus you have the fact that there simply
aren't that many air breathing engines to choose from if you expect them to
operate over a wide range of speeds. Most air breathing engines are
optimized for subsonic cruise, with a few optimized for supersonic cruise
(e.g. SR-71, Concorde, and some modern jet fighters).

So exactly why would anyone prefer a costly, complex, heavy air breathing
engine for rapid acceleration of a launch vehicle, when you can use a more
simple, lighter, cheaper, LOX/kerosene first stage engine instead?

I proposed some time ago in sci.space.tech a solution for a "zeroth"
stage that would boost the rocket up maybe 150 m/s or so and radically
lessen gravity losses.
The usefulness of a zeroth stage is discussed he
http://ambivalentengineer.blogspot.c...-to-orbit.html
A 300 m/s boost would allow stretching the tanks (same engines) and
delivering about thrice the two-stage payload to orbit. A small boost
might not seem much, but if the rocket has a thrust to weight initially
close to 1, it doesn't accelerate much but it travels up at that
300 m/s still and that lessens gravity losses a lot.

My zeroth stage would consist of helicopter rotors getting their
power from the rocket turbopump. The fuel consumption would only
be a few percent of the full-throttle operation (that much is diverted
typically to the gas generator), and not much additional hardware
would be needed. Just a gearbox and a clutch. It's been hard
to get gas generator power levels, but some, when compared to
helicopters seem to be just about having enough power to lift the
rocket, at least at low speed.

The idea is that using the stationary air as reaction mass
is extremely effective at low speeds. It's more of a hindrance
at mach 1 and above but the stage can be discarded long before and
the rocket can operate conventionally. The rotor assembly is easy
to be made to descend at very low speed and be recovered for
reuse.

Note that this is NOT a tip-rocket rotor like Roton or Hiller.
There are no difficult fuel columns inside the fast spinning
rotor(s). Just somewhat ordinary helicopter rotors. I guess they'd
have to be special to operate at such wide speed regime along the
rotation axis (but they gimbal anyway), and I don't know what
efficiency can be gained, I'm totally clueless about the
aerodynamics here. You'd have to have lots of blades to limit
tip speed at least.

Turbopumps and combustion chambers ain't cheap, even if tanks
were, and with this system one can keep them at existing
size and still boost the performance tens of percents, maybe
close to 100%. It might still not be commercially viable,
but it would be cool to talk about the idea's technical
feasibility.