This is not my experience.
I can't speak for all "big instances," but my experience on mastodon.social is that sometimes(!) a post has even been removed before I can post my report, as s.o. ele has already reported it before me. 😀
Other than that, yes, we all need to report posts that are illegal or in breach of our instance's rules.
We might need more #moderators in the future, but as #Reddit has shown, this is feasible.