To me initiative matters the most, the rest can come later.

I go through an insane amount of purifications to land an orange attribute on initiative before locking anything else. I made a mistake and started locking on other orange attributes before landing on an initiative, and this is where I went wrong. Wasted more than I ever hoped for locking and re-rolling. This is my strat if you're in no rush.

If you're in a rush for quick power and stats, you can either keep rerolling till you land on an orange, lock it and reroll until the other stats are blue/purple then move onto the next equipment. (This is what I believe to be the most common method but it will cost you dearly in the long-run if you seek an orange or even purple initiative)

Another lesser and cheaper way is to reroll and settle for a purple initiative then work from there.

I've done this for my light main, all 5 tools have gold initiative. To get the most stats out of the gold purification, i've taken all 5 gears to level 8 refines (all free from events and sage) and it's sitting on a 4k initiative.