I don’t use that as a hard-and-fast rule, but generally my solving does go criss-cross like that, from the NW does to the SE, rather than doing all the acrosses first, then then downs. I find that the fastest way to complete a puzzle.
I always do the Mini - my challenge is to solve it in less than 60 seconds. Today was 58 seconds for a puzzle that had 16 clues (just about maximum for a Mini) so, I patted myself on the back for that.
Nicely done!