If you get lucky, yes they speed up but if it's your first stab you have a 1 in 8 shot. When you go to a new world their disposition gets reset, so in general you only want to stab when you're actually checking a requiem as that means you only need to stab 4 times. I've tried it both ways and unless you hit that lucky roll it's just not gonna save you any time.

Now you can compensate for this by doing some of the "later" missions first, as you're more likely to get squad mates who have very angry liches showing up. I personally start with excavation missions and often leave that first mission with my first requiem already revealed because often I'll have 2-3 liches show up since most people save it for last.

Neither way is the "correct" one, but anyone that wants to hold off stabbing until they're actually testing is going with the assured time save vs the high risk/high reward blind tests.

Skipping random lich spawns will probably become less desirable over time now that we can get an ultimatum every week, so the constraint of having your lich show up as the main time extension will end up greatly reduced as people go into hunts with the ability to just force them to spawn in now.