Any gas cloud of sufficient mass would eventually collapse into a single spherical object. A star, if large enough to support fusion in the core, or a planet if not large enough. This ‘internal pressure drives the gas apart’ theory doesn’t hold water – if this was true, all the gas giants would instantly burst out into a diffuse cloud of gas, and the sun would be even worse, driven apart by gas pressure and the energy of fusion in the core. If the gas cloud has enough collective mass to form a planet or star, that collective mass is enough to overcome gas pressure.

Now, gas pressure would still push back as the gas tries to condense around the center of mass, so it wouldn’t collapse instantly. And if the system has any angular momentum, it will form a disk first, which would then slowly collapse into a spherical shape due to friction/gravitational effects gradually slowing the speed of rotation and causing gas to fall inward.

Where your theory might have some merit is when it comes to planets forming around a new star. If the cloud of gas is more or less homogeneous, why wouldn’t all of it eventually collapse into the center? But if some ‘seed’ nodule of denser material enters into orbit through the cloud, that object would then start accumulating gas around itself, which then increases its mass and draws in more gas, and so on. Everything outside that object’s orbit around the proto-star would tend to collide and be ‘eaten’ by the proto-planet before it could fall inward far enough to reach the star itself.

So, in other words, I think you may have something when it comes to planet formation, but not star formation. A cloud of hydrogen/helium gas doesn’t need any ‘core’ to gather around in order to form a star – it will just gather around the collective center of mass. We already know the early universe contained only hydrogen, helium, and a tiny bit of lithium. The very first stars, at least, had to form without any dense core to start with. But dense ‘seed’ cores might be required in order to form planets around a star in cases when they all form together. (Though ‘rogue planets’ far away from any star’s influence could still form without needing any sort of denser core. And those rogue planets could later be captured into orbit around a star, so it could still be possible to find a core-less planet in orbit around a star.)

IIRC we don’t really know exactly what catalyzes a nebula to collapse into a star and start an accretion disk, but static electricity in the hydrogen cloud probably also has something to do with. Once the protostar starts to form the cloud of dust around it starts orbiting all in the same direction and collapsing into a plane. This is why all planets in a solar system orbit the same direction and are roughly on the same plane. Then the dust slowly accretes into larger and larger rocks. It is believed that gas planets form on the outside because near the sun hydrogen cannot become a solid and begin to accrete. Rather it remains a gas and gets blown out to the outer solar system where it becomes a solid and can accrete into planets. Heavy elements like iron and silicon and magnesium stay solid near the sun which is how so much of it ended up on planets nearer to the sun like earth and mars.