"Why was everything always good, peaceful and stable in "indigenous environments" until "the colonizers " show up?"

One thing that may help resolve this in your head is that invasive species taking over is the exception, not the rule. The vast majority of the time, when a species is taken out of its native environment and dropped somewhere else, for whatever reason, it dies. Maybe immediately, maybe a few generations get off before the population drops to zero, maybe they do great until they're slugged by winter or the rainy season, most of the time the "home team" will kill the visitor without so much as metaphorically noticing.

However every once in a while the stars align and the new species fits into a slot the home team didn't "realize" existed, or they hammer a weakness that the rest of the ecosystem had just been coevolving around for a long time, and you get the invasive species. It would feel like winning the lottery, except that the invasive species then get to grow exponentially and loom very large in our minds and our experiences. They are, despite that, the rare exceptions and not the rule. The rule is that a species dropped into another full ecosystem with no coevolved slot for them just dies.