How important is class rank for admission?

I'd say class rank is quite important because colleges love to evaluate applications "in context" and class rank gives them a lot of context. For example, say that all your grades are only between 80 and 85 and your GPA is much lower than the traditional applicant but you're still always at the top of your class and its just that teachers in your school grade harshly. Then providing a class rank helps a great deal because it helps show colleges that grading is harsh at your school. Similarly, in cases with grade inflation, getting 95+ in all classes may make it seem like you are at the top of your class, but providing a class rank of say 20/150 shows that your school most likely has grade inflation.

However, at the same time, providing class rank is not always essential, you will be fine without it, I got into UCLA without it. Alternatively, you could request teachers to provide this information in their LORs (which is what I did because my school has a strict no rank policy and I got into Brown with this).

It isn't a deal breaker but certainly helps you a lot and colleges do tend to appreciate providing class rank.