It's had very little influence. People who are critical of the war and the Bush administration see the findings of confirmation of what they know. Those who supported and still support the actions will disregard, minimize or ignore them as much as possible.
The main problem is that the US media, given changes in ownership laws and general regulations over the past several decades, was a major part of the problem in the lead-up to the war and dissemination of phony information on Iraq, but also would be a necessary player in bringing prominence to these findings.
When broadcast networks were still the dominant force, and before Reagan essentially gutted enforcement of public service and "fairness doctrine" requirements for broadcast licenses (the airwaves were considered part of the publicly-owned commons, and radio and TV stations were ALLOWED use of them, which meant they had to meet certain requirements).
With ownership and consolidation rules relaxed, starting with Reagan and continuing until today, and the public service requirement gone, corporate ownership of news networks by media conglomerates has changed the very nature of news reporting. Network news used to be targeted at informing the public, as part of this service, and networks did not look to the news as a source of profit.
With media conglomerates now looking to maximize profits from all aspects of their holdings, we see that budgets are cut for all the hard, serious, background investigative reporting, and a stronger emphasis on a news product that brings more ratings, and garners more advertising revenue.
You now see more news stories being influenced by advertisers, you see "sponsored" news stories that are generated often by PR or marketing agencies to give a good view of their clients, and sold as content for news stations to deliver, without disclaimers about their sources. You see a fundamental dumbing-down of the news, and shift towards simple news that focuses on drama or conflict instead of deeper discussion of complex issues.
You see news networks that are loathe to broadcast news that their viewers might not like, for fear of losing ratings. This makes pandering a more likely editorial choice than hard truths. News, for the most part in the USA, is no longer about informing. That is why so many now refer to it as "infotainment."
Link this all back to the run-up to the Iraq war, you had a media that did not ask critical questions, basically cheerleaded the chest-thumping that was going on, and was manipulated and duped into propagating misinformation to the public.
Any retrospective looks have and/or will gloss that over, since it will be the very same media reporting about it. Any serious examination of the Chilcot findings would have to be paired with an examination of the messaging machine, processes and failings, and would also lead to a very uncomfortable message that our desire to fight that war, and the consequences, were, in a word, based on falsehoods that we willingly accepted, with horrible consequences for the region and our long-term security (ISIS, anyone?). That would mean reporting news that would be critical of the reporting media organizations, and also their viewers, the US citizens. Since they don't want to tell people news they don't want to hear, that's not going to happen.