In the 2nd friedmann equation, the pressure term of dark energy is what counteracts the energy density and cause acceleration of the expansion of universe. But normal pressure gradient arises from the difference in pressure between 2 regions, if the pressure is the same between 2 regions then pressure has no effect. Yet the universe is quite uniform in "pressure", so where does this pressure gradient come from to propel expansion?
It is tempting to imagine the universe as a ball of gas. A positive pressure would make the ball expand while a negative pressure would make the ball contract, and as you say it is the pressure difference between the inside and outside of the ball that drives the expansion or contraction.
But this simple idea does not apply to the universe. The effect of pressure in general relativity is more complicated than you think. In particular a positive pressure makes the universe contract while a negative pressure makes it expand. This is exactly the opposite of the way your ball of gas behaves.
I go into the reasons for this in my answer to Intuitive understanding of the elements in the stress-energy tensor. I'll make a simple analogy to try and explain what is going on, but be aware this is only an analogy so be cautious about taking this too literally:
A positive pressure, as in the ball of gas, is produced by the gas molecules whizzing around and colliding with things. So the positive pressure is related to the kinetic energy of the gas molecules. But energy gravitates in the same way that mass does, so a positive pressure means a positive energy and that means there is a gravitational attraction. In effect the positive pressure increases the energy density so it tends to make the universe contract.
(The reason you need to be cautious about this analogy is that the negative pressure due to dark energy does not mean it has a negative energy - dark energy has a positive energy density and the negative pressure is a result of its equation of state.)
So the answer to your question is that pressure does not affect the expansion of the universe in the way you think it might. That's why the existence or otherwise of a pressure gradient is irrelevant.