Not sure why there is still an open question here. Let me try to clarify.

EA's story progression - Will cull (emigrate) sims whenever it feels like it. This can and will also happen when story progression is off and the game goes into emergency overpopulated or unbalanced corrections even when it doesn't need to.

Shimrod's mod that I linked to is supposed to block that kind of emigration, specifically targeting EA control (not NRaas or AwesomeMod).

NRaas StoryProgression - Emigration is disabled by default. It can be switched back on, this is actually not easy to do and it's not easy on purpose because so many players hate surprises, but emigration will only happen under a few very specific sets of circumstances. The player has total control.

AwesomeMod StoryMode - Individual households can be marked to never be emigrated. I don't remember the rest of how this works, to be honest, but I don't recall having any surprise emigration at all many years back when I was an AM user.

There shouldn't be constant career changes, but the mod will in the beginning of its usage attempt to put sims into careers that are more in line with their traits and LTWs (Lifetime Wishes) than the random career they may have picked up by way of EA or for premade sims, whatever it is they had as an originally defined career. That can make for a flurry of career changes seemingly all at once, but then things should settle down pretty quickly, with only an occasional career change. Of course sims newly arriving to the world and newly aged up YAs are going to get jobs (as well as Teens picking up part time jobs along the way), Elders are going to retire, etc. Sometimes sims get fired for poor performance, then they will also be pushed to pick up a new career. Sometimes other sims in town just need some co-workers and bosses for their own advancement to happen.

This isn't Free Will or sims doing stuff just because they randomly feel like it, this is SP pushing sims into what it "thinks" are logical paths for them to follow.

If you don't want sims changing jobs at all without your involvement, then SP > Town (Caste, Household, Sim) Options > Allow Find Job > False. If you don't mind a flurry of activity all at once and then only the occasional career change but just don't want the notifications, then the stories can be suppressed. These things will still happen if stories are suppressed, just without the notifications on-screen.

I don't know how the discussion shifted to Oasis Landing, but that is not a homeworld. If we want to run SP's progression on the three WA worlds, Uni, and/or Oasis (not everyone wants to), that's fine but we do have to take into account that there are limited to no career rabbitholes. Place a rabbithole diner or hospital in one of these worlds and yes, under default settings the local population would be expected to flock there to get jobs because SP doesn't have anywhere else to push them. These worlds require special handling, at least in terms of careers, and I would think just switching off Career: Allow Progression (which also switches off Allow Find Job) at the Town Options level would be a good way to handle that until we decide what we want to see happen in these worlds.

SP doesn't push street performer assignments, it's actually Showtime itself that does that. SP can't prevent it entirely in any world, but disallowing the three performance careers and the three performance career skills (SP Skill module required) can at least slow down the unwanted assignments.

Since the local population in Oasis is meant to be fluid anyway, depending on sims visiting it and who their descendants are supposed to be, you may want to just switch progression off entirely there unless you had some specific reason for wanting to run it.