1) I fully agree that air-breathing engines are heavier per unit of thrust. Merlins - pretty good rocket engines by that parameter - lift, as you say, 150 times their weight; for air-breathing engines it's 5-10.
2) Aircraft engines usually don't need to be particularly thrusty; wings generate lift, so you can take off even with T/W < 1.
3) Liquid oxygen has density about 1.1 that of water. Kerosene, in comparison, is about 0.8. Difference less than 1.5 times, but the mass ratio is more like 2.5:1, so ditching oxygen tanks - especially at the start, when the system is relatively heavy - can save quite a bit. And why engines' T/W is relevant here?
3) My comment was to compare liquid oxygen with liquid hydrogen which is 0.07 of water density, so you get 16 kg of oxygen for 1 kg of hydrogen for the same tank volume. Skylon uses loads of hydrogen. It's those hydrogen tanks that make it so big. Such tanks are also a lot of dead mass to carry to orbit... Also costs probably scale with dry mass, not wet mass.
1) Skylon has air breathing engines. Air at sea level is about one thousandth the density of normal rocket propellants and thus the machinery for compressing that air (inlets, spinning compressors, in Skylon's case heat exchangers) is large. Air breathing engines need to be much larger than pure rocket engines of equivalent thrust, and thus heavy.
2) Since it's the first stage engines, they are large because of that. SpaceX for example has nine engines in the first stage while only one engine in the second stage. If we assume 500 kg per engine, 8 engines saves 4 tons. Since their payload is about 10 tons, they would only have about 6 tons of payload if they flew with all engines to orbit. (Not a very precise calculation I admit.) Why would they do that? And that's with rocket engines that have 150:1 thrust to weight ratio. Modern operational air breathing engines have about 10:1 thrust to weight, one fifteenth of Merlin 1D.
3) Carrying oxygen isn't as big a problem as it seems to be. It's denser than water so the tanks are actually small. And rockets already have tremendous thrust to weight as we have demonstrated.