The left is also fond of reminding us that we are all descendants of immigrants...and we have absolute power, so the little power we had at first, which was negligible, grew as our numbers did. That concept still holds. Politicians pandering to the Hispanic community, going so far as to try and speak Spanish in an election debate, would argue against your point. These immigrants and their illegal cousins have a lot of power, so much so that one party is banking on their numbers increasing for the express purpose of establishing a permanent Democrat voting block.
The first settlers had no power at first. It was a long slow process that pushed the indigenous peoples out of their lands. It doesn't have to be government sponsored. The Americas weren't government sponsored. Settlers had to pay their own way here and make it however they could. Everyone came for their own purposes. People from all over Europe found their way here independent of government sponsorship. Britain claimed the land, but that doesn't speak to the peoples' intents, and the Revolutionary War was fought to bring that point home.
Also, immigration to the US has never stopped, only for some reason, the left wants to make a break in the chain and call everyone before a certain date, "colonizers" or "colonialists" as if, one day, everything changed. My point is this. One side can't simply call everyone immigrants and descendants of immigrants while using the term "colonialists" to describe one race, when everyone was coming here, and still is, for the purpose of setting up homes and moving families over.