Is it better to complete the stability in an echo or should I move on to another one after completing the main objective(s)? TIA
You should kill all mobs that are on your way to the objective and after finishing the objective if there is still a big mob group around definitely clear them out as well.
Then you should move on to the next echo.
Filling up bonus stability is not really worth it for the most part. There are some echo layouts where you already fill like half of your bonus stability just by killing everything on the way to the objective, thats more than enough.
Think of it this way: you get a guaranteed minimum stability for the echo. You will already have some bonus stability from kills on the way. So if you just leave and start a new one, not only will you fill stability faster but you will also get more echo rewards overall. So it’s always better to just rush the objective.