Cold gas thrusters, per their description have some hard performance limits tied to the speed of sound for the gas that cannot be optimized while still being a cold gas engine.
Nitrogen has a speed around 300 m/s and ISP of 72 seconds, lofting a 1000kg gets maths suggesting using most of earths atmosphere as propellant, assuming a magic zero weight tank to put it all in (putting the atmosphere inside the rocket would at least reduce drag)
moving to Hydrogen with it's higher speed of sound you go from an ISP of 65 to 272 from a speed of sound of 1320 m/s. This is not terrible, and some real world rockets had this sort of performance, though the very low density of hydrogen makes storing enough of it in a small and light rocket tricky. For comparison the space shuttle RL10 has an ISP of 465 from an exhaust velocity of 4500 m/s.
Throwing some very generous numbers around, if we take the space shuttle external tank and swap the LOX volume for liquid hydrogen, we get a tank holding 140 tonnes of liquid hydrogen, with an empty mass of 26 tonnes. Dump that into a calculator with an ISP of 272 and we get a total Delta V of around 5 kilometers a second, which is a quite reasonable first stage, but this is with a payload of zero so no second stage.
We could shave some tank mass down, but we also should add mass for whatever is turning the liquid hydrogen into gas that we can use for our cold gas engine. And the easiest way to do that is burn it with oxygen, and if doing that we might as well burn more to add thermal energy... and now we have a normal hydrolox rocket not a cold gas system.
Trying to store the hydrogen as gas will end up with very heavy tanks, hence why existing rockets use cryogenic hydrogen even with the issues it brings.
There are other ways to boost performance from a 'cold gas' Nuclear thermal rockets add thermal energy to hydrogen, making them 'hot gas' rockets, and various ion and plasma engines bypass the speed of sound physics by using magnetic or electrostatics forces to accelerate the reaction mass and get four digit ISP from exhaust velocities where relativity starts to matter. Neither of these are good options for launch from earth, hence the continued use of chemical rockets.
So cold gas thrusters are pretty much the worse possible thing to do with reaction mass in a rocket, and are not going to get anything to orbit from Earth. They generally only make sense when safety or simplicity concerns mean show up for things like reaction control or manned maneuvering systems.