This is more complicated than it seems. Yes, current models do take radiation into account. But it’s not currently significant. And even if it were, it wouldn’t have the effect you think it would.
Pressure will only push things apart when there’s something to push against, and with the expansion of the Universe, we’re talking about expansion of space itself. You can’t push on space-- The only way you can affect it is gravitationally. But pressure is still significant. We think of gravity as being produced by mass, but that’s only an approximation. In actuality, gravity is produced by mass and pressure (and also by other sorts of stress, and also by momentum flux, but let’s not worry about those here). For most things, the pressure (in appropriate units) is far less than the mass density, so we can ignore it, but for radiation (light and anything else moving at extreme speed), the pressure is just as great as the density. So the positive pressure of light would actually tend to cause the Universe to contract more, or at least slow down in its expansion.
The significance of various sorts of “stuff” varies over the history of the Universe. As the Universe expands, both radiation and normal matter become less significant, but radiation drops in significance more quickly than matter. So now, matter is many orders of magnitude more significant than radiation, but in the very early Universe, radiation was more significant.
But there’s yet another important type of “stuff”, that being the cosmological constant (AKA dark energy, AKA quintessence). The cosmological constant can be thought of as a substance with positive density, but negative pressure, with pressure nearly as large as its density. Moreover, while matter and radiation both become less significant as the Universe expands, the effect of the cosmological constant does not change. The Universe has currently expanded to the point that matter is actually less significant than the Cosmological Constant, which means that the current evolution of the Universe is dominated by that negative pressure. Negative pressure produces negative gravity, which is what’s causing the current expansion to accelerate.
Yes, I am quite aware that it’s counterintuitive that negative pressure should encourage expansion. Unfortunately, I don’t know of an easy, intuitive way to explain why this is so.
