My understanding, which to be honest, has been gleaned totally from social media, so isn't likely to be a fair representation, is that the issue "the people" have with this incident, is that Willoughby and Schofield lied.
They were seen (and video evidence exists) to show them walking past the queue of people, and into the building. They were also seen (and again, video evidence apparently exists) to show them walking past the coffin.
When this was raised and the question asked "why should they have been fast tracked", they said that they didn't jump the queue and that they didn't file past the coffin, but were there "working". They said they were only standing on the media platform at one end of the chamber.
The "story" grew legs and ran away, with social media imploding with indignation, resulting in ITV issuing a statement saying that Willoughby and Scholfield were asked to film a piece to be shown in their programme, and as such were entitled to use the Press fast-track process.
Unfortunately, there was no filming allowed in the chamber where the coffin lay, and in any event, they were not accompanied by any film crew. The only footage which has been seen, is what appeared to be a "selfie" video, shot on a mobile phone, of Willoughby and Scholfield after they had been through the chamber, saying what a "profound moment" it had been. No professional film and sound crew to record the moment.
They said they were filming a piece to be shown on the programme for people who were unable to be there themselves. Unfortunately, they were filmed on the Friday (they apparently - I don't know, as I don't watch it - don't present This Morning on Fridays) and therefore the first opportunity they would have had to show the piece would have been on Tuesday - the day after the funeral, so unlikely to have been of any interest to those who hadn't been able to attend in person. The same people who could have watched the whole thing practically 24/7 since the coffin arrived at Westminster Hall.
The general consensus of opinion appears to be that if Willoughby and Schofield had held their hands up and said "yes, we did go and view the coffin and we did bypass the queue, and we're really sorry, we misjudged this totally" then the whole thing would have died down as quicky as it came to life.
And this is what happens when you have a week off work, with nothing to do - I ended up getting sucked into the increasingly hysterical posts on Twitter, Facebook, Mumsnet and TikTok. I feel I know far more than is healthy about this non-story.