Having played several hundred hours of KSP I can confirm you shouldn't underestimate momentum. It's amazing that it's even possible. I think the number I've heard is if the earth's gravity was something like 10% stronger this wouldn't be feasible.
Fuel is heavy, and most of it is spent on... lifting fuel.
Having a nuclear explosion for example behind the rocket would surely be enough to push it into space... even on Jupiter.
Not quite, it's all about mass flow.
It's not about the amount of thrust. It's about being able to carry enough fuel to sustain that thrust long enough, without the weight of the fuel making the rocket too heavy to lift high enough to reach orbit.