Actually, I think that for most students they hit a wall in Algebra regardless of how they learned basic mathematical operations. This wall has to do with abstraction of principles not with mechanical operation speed.

If what we are trying to teach is this abstraction, then the rote table method is arbitrary* and a time sync. The students will still learn basic sums over time, it just won't be the focus of the practice.

*For instance, my education with the tables went up to 12s. Why in the world did we choose 12? 16 would have certainly helped me out more.