Part of the argument can be reasoned without using any "heavy handed" general-relativity, but it is a long road.
Part 1: Relativistic objects are "pulled toward" gravitational sources more than predicted by newtonian mechanics. Light is pulled twice as much. This needs general relativity, see here.
Part 2 Pressure results from exchange of moving particles. In an ideal gas, the moving particles cause pressure. In a solid of liquid, it's more complex. There is degeneracy pressure and attractive forces that allow object to exist in tension. This is due the the exchange of negative energy virtual photons.
A system under pressure (such as a hot gas) contains moving particles that have "extra" gravitational attraction to our test mass. Conversely, our test mass feels an extra attraction to these particles. Although the kinetic energy of the particles contributes to the total mass and thus the gravity, the pressure creates extra gravity beyond that of the mass and kinetic energy alone.
An example: Consider a thin hollow spherical mirror filled with photons (again, we are working with a weak gravity source). 343
We place a test mass (that does not interact with light) just inside the mirror. Due to the shell theorem we need only consider the tiny amount of mass/energy that closer to the center of the sphere than our test mass. The Newtonian calculation would be G(our_mass)(energy_of_photons)/r^2, but the actual force is twice that due to the pressure. If we are outside the mirror the newtonian formula again applies. Although there is pressure inside, there is tension in the walls of the mirror, in effect it is a a balloon inflated with photons! The pressure and tension terms cancel themselves out. When you are outside of a spherically-symmetric object, only total mass matters and the internal pressures will always cancel.
When pressure is due to gravitational compaction you can't escape hard-core relativity if you want to account for it: On earth, the pressure contribution to gravity in the core is only 1e-9 of the density (mass) contribution. This "tiny" amount of pressure isn't canceled out by tension as in the case of our mirror balloon. However, to understand why it gets canceled out we would need to invoke full-beast-mode-general-relativity because relativistic effects are also 1e-9 as strong as Newtonian gravity for Earth (no it's not a coincidence that both are 1e-9). Pressure is only important, in comparison to density, when P ~ (density)c^2, and that c^2 makes even core-of-Earth pressures look small. 343