"And the whole precooler idea means they can run on atmospheric oxygen for more of the flight profile, which in turn brings about big savings."
The challenge is that reducing oxygen mass in a single stage to orbit vehicle is NOT a big weight savings by the time you get into orbit.
Yes, oxidizers are the largest fraction of a launcher's mass when it's sitting on the ground. But the thing is, most of that mass goes out the tailpipe by the time you reach orbit. Take a moment to consider the rocket equation (if you'll pardon the simplification):
Delta-V = Exhaust Velocity x natural log (fueled mass / dry mass)
The fueled mass doesn't matter much by itself because an SSTO is going to throw most of that away. More important values are the exhaust velocity (or specific impulse times G, if you prefer) and the ratio of fueled mass to dry mass. An SSTO really wants to have a big ratio between fueled and empty masses and, of course, a high exhaust velocity would be nice.
Skylon claims to achieve a specific impulse in the atmosphere, and I won't question that. So let's look at the ratio of dry to fueled masses for SSTOs. The dry mass consists of frame, oxidizer and fuel tankage, avionics, landing gear, engines, cargo, and anything else that didn't get dropped during flight.
In the case of a rocket, the in-orbit leftovers of the oxygen system are oxygen tanks. Rocket oxygen tanks are light, around 1% or less of the mass of the oxygen they carried. For example, the shuttle external tank's oxidizer tank was about 10 tons for 600 tons of liquid oxygen.
There are heavier bits on a rocket headed for orbit than oxygen tanks. The perennial favorite fuel, hydrogen, needs about 10% of its mass for tankage. The shuttle ET's hydrogen tank was about 15-20 tons and held 100 tons of liquid hydrogen. The dominant factors in tankage mass are volume and pressurization - interestingly, liquids' weights scarcely impact rocket fuel tank design, even when you're looking at the weight at 3Gs. Liquid hydrogen has low density (1/14th of water, 1/16th of liquid oxygen), and its fuel pumps need high tank pressurization to help prime them, on the order of 35psi. That calls for a large, sturdy tank and thus high dead mass per unit of liquid hydrogen.
Engines may be a large part of an SSTO's mass, or not. Dense fuel rockets, like kerosene-oxygen rockets, achieve better than 100:1 thrust to weight ratios, meaning a 1000-ton SSTO requiring a 1.3:1 takeoff thrust only has 13 tons of engines. SpaceX is nosing around 150:1. Hydrogen:oxygen rockets rarely beat 75:1 and tend to be around 40 to 60:1 simply because they need such massive fuel pumps for a given thrust level - fluid pumping horsepower requirements are largely dominated by fluid volume, not mass, but thrust levels are determined by the rate (mass per unit time) you're burning fuel. Dense fuel engines can burn a lot of fuel mass with a light pump; hydrogen/oxygen engines are screwed by hydrogen's low density since they need a big pump to move a little fuel.
Rocket engines aren't the only game in town, as Skylon and scramjet enthusiasts demonstrate. However, airbreathing engines have much worse thrust-to-weight ratios than rocket engines. A good, military, afterburning jet engine manages about 10:1 thrust-to-weight ratio. Scramjets might not break 1:1.
Skylon's SABRE hopes to reach 14:1, which is good for an airbreather. Still, that's a lot of deadweight to carry into orbit just so you can shrink the already-lightweight oxygen tanks.
Skylon's design (and that of other airbreathing SSTOs) runs into other issues. It needs to be sleek and aerodynamic so it can fly at high speeds through the atmosphere. "Sleek and aerodynamic" mean "high surface area for a given volume." That means a proportionally high mass for tankage and frame compared to chunky SSTOs like the Kankoh Maru, SASSTO, or even the VentureStar. The "high speed atmospheric flight" also means "more heat shielding," compounding the surface area issue.
Admittedly, a larger liquid oxygen tank does drive several other weight increases: more heat shielding, larger frame, more engines, etc. However, those are modest increases. Frames, rocket engines, and heat shields are percentages of masses they carry, not multiples.
I'm not saying Skylon's impossible, just that it took the more painful route. The "big savings" in oxygen mass really isn't a big savings since oxygen tanks are so light, and you're getting those oxygen system savings at the expense of enormously heavier engines, more challenging aerospaceframe design, heavier heat shielding, and so on.