A few corrections on rocket engine performance are needed:
First, you can't calculate the thrust of a rocket engine simply by looking at the chamber pressure and the area of the chamber top exposed to it. You would need to integrate the pressure forces over the entire interior surfaces of the chamber
and the exhaust nozzle. And, the exhaust at the end of the nozzle does not have zero static pressure. (That would require an infinitely long diverging exhaust nozzle.) If it is designed to run at a particular altitude, then it will be designed to have that same static pressure as the atmospheric pressure at that altitude. For rockets that start near sea level and go into orbit (or beyond), the exhaust static pressure is usually optimized to produce the largest time integral of the thrust through the powered part of that rocket's designed ascent path. So , at sea level, it is over-expanded, and the static pressure of the atmosphere is more than the static pressure of the exhaust at the end of the diverging nozzle. That pressure difference creates a diagonal shock wave to compress the exhaust gases as soon as they get out of the nozzle, which ends up over-compressing the exhaust gases once that conical shock meets iteslf at a point in the center of the exhaust stream and progesses back out to the edge. At that distance from the nozzle, the shock reflects back into the exhaust gases as an expansion fan (spreading group of waves) and when that gets to the center, it then over-expands the exhaust gases again as it progresses back out to the edge, where it makes the exhaust gas static pressure again lower than ambient atmospheric pressure. So the compression shock/ expansion fan pattern repeats, and what we see it that familiar diamond shock wave pattern in the exhaust gases. If you look at a rocket exhaust in space, you will note that this diamond shock pattern is not occuring in the "vacuum of space".
Next, while it is correct that a rocket and its propellant in free space will always maintain the same postion for their combined center of gravity, it is not so simple as saying that
As the rocket arrives at a star 4 light years away, the exhaust gasses are located 4 light years in the opposite direction.
The exhaust gases will be moving at different velocities with respect to the center of mass, depending on when they were expelled from the moving rocket vehicle. Initially, the first gas expelled will be traveling much faster than the rocket, so it will move father from the center of mass than the rocket (and remaining propellant it is still carrying) than the rocket does in a period of time. Unless that rocket ultimately exceeds the relative exhaust velocity of its gases, the
first exhaust gases released will always be farther away from the center of mass than the rocket propelled vehicle that released them. And, the last released propellant gas might not be moving at all with respect to the center of gravity if the final speed of the rocket just reaches its exhaust gas
relative velocity. Or, if the rocket's velocity with respect to the center of mass does eventually exceed its propellant relative exhaust velocity, then the last exhaust gas will actually be moving in the same direction as the rocket away from the center of mass.
Finally, I should point out that planets that are more massive than Earth may have atmospheres that are more dense and deeper that what we have here on Earth. If so, then rockets launced from the surface need to expend more propellant to overcome the higher atmospheric drag, and do so longer to get higher to reach the top of the atmosphere, and do those things while attaining an even higher orbital velocity due to the heavier planet. So, there are multiple compounding effects that need to be taken into account.
And, if the planet's atmosphere is always cloudy enough, intelligent life on its surface may not have seen the stars, at least not for a lot longer in their evolution than us humans.