You don't have to final hit them.

Freezing them a few seconds before they die, either by your bullets or your fireteam's, counts.

Did both strikes quest in 3 strikes, i'm on my 6th crucible match and i'm about 30% done.
Some objectives, like killing 18 slowed guardians and picking up bits left behind are nigh impossible. The only ability that slows on Warlocks is the grenade.

So you must kill 18 people within the grenade's field, but before they get frozen, and then you must run to their corpse and grab the thing before either it disappears or you get shot. It also implies running away from objectives every time you get that kind of kill for it to even count.

I can't say for Gambit as i haven't tried those yet, but Crucible objectives are objectively harder than Strikes, and also disadvantages certain classes specifically.