I see you still failed to address the question.
Why?
What force causes the gas to leave the rocket and what other body is involved?
The only options are to reject physics or accept rockets work in space.
So, let's see: my 'vessel' (i.e. the midget soldier and his machine gun & ammunition) is :
About 4.5 X superior (more efficient) in terms of mass expelled per second / vs vessel weight
About 3.3 X times inferior (slower) in terms of 'muzzle / exhaust exit velocity'.
You mean the Ariane is 3.3 times faster.
As a simple approach you can use the rocket thrust equation and ignore the pressure term.
So just taking your numbers for the machine gun, the force will be 1.2 kg/s * 825 m/s = 990 N.
Assuming it is going straight up, then the force it needs to provide just needs to counter gravity, and thus for a 100 kg load needs to be roughly 980 N.
But quickly checking, your numbers are off. The bullet weight isn't 24 g. That is the weight of the entire round, including the gas and cartridge which would be ejected at much slower speeds.
The actual bullet, which is what leaves at the quoted speed is only 9.6 g, or 0.4 times the mass you used. That means the thrust would be roughly 0.4 times the previously calculated thrust or 396 N.
That isn't even enough to lift your 50 kg person.
If you have a correct percentage you can also simplify it a bit.
If the percentage mass flow rate multiplied by the velocity is greater than g (roughly 9.8 m/s^2), it can fly.
Sticking in the original numbers you gave for the gun person that gives 9.9. Correcting it to the actual bullet (0.48%) you only get 3.96. So you aren't going to fly.
As for the Ariane, that is quite a bit more complex. It has 2 boosters and a core, each throwing out some mass at some velocity. And there are different models. I hate it when companies do that.
The 2 boosters give a collective 3429 kg/s mass flow rate.
The core gives 315 kg/s.
So that is already quite different to what you have said. That is 3744 kg/s, or 0.49% of the mass of the rocket.
So when both numbers are corrected, you end up with the Ariane ejecting a slightly larger fraction of its mass per second than the gun is (at least when focusing on the quickly moving ejected parts).
So in reality your comparison would be:
Ariane is about 1x the fractional mass flow rate.
Ariane is about 3x the velocity.
But I can't find the numbers for the velocity anywhere.
But if I just take your word for it, then we end up with a thrust of 10292256 N, enough to lift a ~ 1 million kg object.
The option is to reject explanations that are not based on genuine physics but bullied into the psyche of the masses.
No, the physics being discussed are based firmly upon reality and confirmed by mountains of evidence.
You not liking these laws of physics doesn't mean they aren't real.
So the option remain the same:
Reject physics, or accept rockets work in space.