You don't need composite tanks for SSTO. There have been normal rocket stages that have had high enough mass ratio and specific impulse big enough for SSTO, like in Titan, but they weren't used as SSTO.
One of the secrets is dense propellants which increases mass ratio considerably. It also makes higher thrust to weight ratio engines easier to implement (because pumps primarily pump volume rather than mass), and with identical initial acceleration, the dense propellant higher mass ratio lower impulse vehicle gets lighter faster and thus has higher acceleration towards the end of the flight, meaning less gravity losses. (Also air drag is less with a smaller vehicle, but it's not a big effect, though it is proportionally bigger for smaller vehicles.)
Hydrogen has the best specific impulse but it is non-dense. It is also hard cryogenic, meaning potential complexity in the vehicle, ground infrastructure and operations.
Also, big vehicles might be more expensive, even if their contents (the propellants) don't weigh much.