There are several doomsday movies that predate The Road, were better at the time, and hold up far better today, IMO. Children of Men (2005) would be my top suggestion. The rare movie I've gone back and watched multiple times. You know why it's good? Because the central premise didn't come true (humanity is infertile), but we live in that dystopia now anyway. It's like Black Mirror but for the police state. It didn't go into hyperbole like The Road or The Day After Tomorrow. It was a realistic, slow-motion apocalypse. Going out with a whimper instead of a bang.
However, Children of Men is admittedly not as bleak as The Road, so not a fair comparison. If you really want a film that produces bottomless despair in the viewer, I recommend Threads (1984) instead. It was produced by the BBC on a shoestring budget, and yet it's still a better film. I can't say I recommend it, but if you do watch it, the ending is on par with Requiem for a Dream for being memorable and filled with doom and horror.
The Road just wasn't very good IMO. No Country for Old Men, on the other hand? Same writer, different genre, but excellent.