As sometime who's memorized but also forgotten their timetables while still managing to go through Algebra I. I don't understand the importance for memorizing times tables (or any other tables for that matter).
I once had a roommate who I encouraged to divide our grocery bill by 2 in his head for splitting the bill, by the end of the year he was quick and accurate.
You can definitely get past algebra without knowing your times tables, but algebra isn't going to help you in the checkout line to make sure you received the correct amount of change and bought the correct number of goods.
practice makes perfect, exercise your brain like your body