Cats could have perfectly round irises like most other animals – there’s no fundamental reason for the mechanism to be different – round pupils would work just as well.
I’ve long felt that the reason has to do with the optics of the situation. If your aperture is defined by a circular region, then shutting down that aperture to smaller sizes restricts the amount of light coming in at the cost of losing some spatial resolution along all axes equally. This is how our eyes work.
If, instead, you close to a slit, you can restrict the amount of light entering as before, but you retain the same spatial resolution along the vertical axis, at the expense of losing more along the horizontal axis. That’s a significant difference.
So when the light gets bright, we lose resolution along both axes. A horizontal fine line, like, say., the tail of a mouse, gets lost the same as vertical fine lines, like blades of grass. It doesn’t matter so much to us – we’re above the grass looking down at the mouse. But to a cat, the mouse tail is still resolved, still seen against the background. The cat can’t resolve the individual blades of grass so well, but that doesn’t matter – the interesting, edible stuff is horizontal, and that’s still resolved, without the confusing vertical stuff you don’t need anyway/. And the cat is looking through the grass, not down.
I note that snakes have vertical slit pupils, too. They have the same advantage of being able to see fine horizontal lines, too.
i say 'mouse tails" here to make it clear why this might be an advantage to cats and snakes, but it’ll be true for any mostly horizontal features – bird tails, insect bodies – that stand out from the general run of vertical “clutter”.
I note, as the OP does, that large cats don’t have slit pupils. Look closely at a lion or tiger and you’ll see this is true. And I note that lion prey doesn’t necessarily have a lot of horizontal lines to it. I strongly suspect that T. Rex didn’t have slit pupils, as he is often represented. I suspect they had round pupils, as shown in Jurassic Park.