"The greater good" is a bit abstract, and your framing suggests it's somehow separate from "the people who live there". A different framing of this question is should individuals be able to degrade ecosystem wealth in order to maximize their personal wealth?
This is the climate story in microcosm. We all know burning carbon is against the "greater good", but if we can pretend that our high-energy lifestyle is somehow independent and unconnected to the planetary ecological systems that support us, then of course, why shouldn't I mortgage my descendants' future for some toys today.
It's the United States, that's virtually a constitutional right. If your skin color and cultural conformance checks out.
Eradicating predators created a very convenient, intermittent period where this was less of a necessity, but it also had quite negative externalities.
So the question isn't "is the greater good more important than the people who live there?", but more "is the greater good more important than the convenience of some people who live there?"
This is a question we have to ask ourselves a lot; nobody wants to live near a landfill, a prison, a sewage treatment plant etc, and yet we want them for the greater good.
It's not just predators vs livestock, there are a bunch of things that we didn't used to understand about ecosystems that we do now. The societal cost of letting people displace or kill wild animals with abandon is quite high.
Zoonotic diseases are the first to come to mind. How many preventable cases of Plague or Lyme disease is that livestock worth to us?
https://cpw.widencollective.com/assets/share/asset/pzqhipzb1... (see Funding)
Cursory search shows that not just Colorado does this.
Well if they liked it, their anscestors wouldn't have killed the wolves in the first place.
"is the greater good more important than the original people who lived there?"
It is also worth noting that not every good that is a "common good" is a necessary good. There are many common goods that can be legitimately given up because there is a hierarchy of goods as well.