Then why did Cosmic Fury even bother with giving an explanation for aliens on Earth at all? The simplest answer is that it knew it had a chance with the story it was telling and went for it. Cosmic Fury, and Dino Fury before it, have made a point to find engaging ways of back filling old continuity errors in the franchise. Dino Fury did a whole courtroom drama with it! The S.P.D. timeline issue is something that writers were aware of and took the opportunity to address,
Could Billy have gone more in-depth that actually there were some aliens on Earth already just to satisfy anyone who remembered “Sam Part 2”? Possibly, but Power Rangers Cosmic Fury wisely focuses on its own unique story and only adds little references and explanations in a way that organically makes sense. As much as hardcore fans would want the longer explanation, it would kill the pacing of the show for anyone else watching.
There’s also a key difference with Cosmic Fury that most other seasons have not had. Cosmic Fury looks to be the last Power Rangers season for awhile. There’s a Netflix reboot in development and it’s extremely unlikely it’ll even address the TV shows continuity. There’s a solid chance that Cosmic Fury could be the end of 30 years of continuity.
This means the people behind Cosmic Fury knew there probably wouldn’t be another season that would have to worry about aliens suddenly living on Earth. With no upcoming season they could finally shift the status quo of Power Rangers continuity to what S.P.D. had set up without alienating new audiences. This was the only chance and they took it.
So yes, the smaller details don’t match up, but it mostly works. If you really want to make it work, perhaps a few more aliens came to Earth, lived with Piggy, and had a settlement of sorts that flew under the radar. Z did live most of her life on the streets, it’s not impossible she fell into a small alien community.
And like previously stated, Power Rangers continuity has almost always had major issues. Lost Galaxy claimed it was set “sometime in the not too distant future” after the previous season of In Space and in that time Earth had constructed a giant space colony that was meant to cross the galaxy. Yet later seasons made it so there was only a gap of about a year between In Space and Lost Galaxy. It’s not impossible Earth was able to construct a giant space worthy colony in a year but it’s pretty unlikely.