What would happen if we added this rule to red-black trees?
Yes it stays $O(\log n)$. Compared to the proof of the original one, we have no three red consecutive on a path from the root to a leaf. The bound on the blacks stays the same in a path. The length of the path is now at most three times the number of blacks which is at most $3O(\log n)=O(\log n)$.

The insertion changes indeed. Now we have strictly more cases, since we are supposed to regard two levels up in the tree after inserting and rotating. However, I think a case distinction suffices as well similar to the one in the original r-b trees.