This. Although said tongue-in-cheek, is a basic principle. Once you understand the concept and the mechanics behind something, what rote learning buys you is to free cycles in the future so that you can take up the next-level task. Skipping this step is like expecting someone to play the piano just by learning how to read a music sheet and where each note is in the keyboard. You build up your muscle memory so that you're able to take on more complex pieces as you progress.
Spending time memorizing the multiplication table, then, is more efficient on both accounts: you will not need to do "a lot of multiplications" to begin with, and it takes less time for your brain to "cache the results".
Maybe that works best for some people, but as a kid I didn't do it. I used a printed multiplication table while tackling some more-interesting problems[0], and let the table soak in as a byproduct. It went quickly and did not turn me off on math for life. Paul Lockhart in Arithmetic also recommended this.
[0]: Multiplying two-digit numbers by one-digit, and that sort of thing. Lockhart had more artistic pursuits in mind.
(An obvious reason this might not apply: I was more talented than my grade-school peers. But most kids would learn arithmetic better if not forced to before they're ready, and then the talent difference would matter less.)