Of course, the family invited Trump. Of course, the family allowed pictures.
It's not up to the family. Politicians are not allowed to hold political events and take pictures for the purposes of campaigning at Arlington. Nobody can make you admit that this was a political event and Trump was taking (very tacky) pictures for the purposes of campaigning. Nevertheless, any reasonable person would agree that's what happened, and that specifically, Trump's plan was to pretend that this was an official, public ceremony and blast Biden and Harris for not attending.* You can refuse to accept this and keep insisting "Nuh uh" all you want, but that doesn't mean you're making a good case or that you're a good debater. It just means that you're being obstinate.
*To wit, here's Fox News trying to frame the story that way once they received their marching orders:
Interestingly enough, this used to have a community note attached to it correctly pointing out that there was no official event, only a private ceremony, but it's no longer there, presumably because Musk intervened and had it removed.
This is why no charges were filed.
Use your head for a moment. Pretend for a moment that you agree with me that Trump was behaving illegally in this case. Do you really,
really, think that any federal prosecutor, a few months before a presidential election in which Trump himself stands a good chance of becoming their boss, would dare risk their career by launching a lengthy, controversial prosecution against a powerful political figure who will undoubtedly take revenge if they're elected, all over a fairly minor offense? That would never happen. I would go so far as to say that Trump could have unzipped his pants and pissed all over the Tomb of the Unknown Soldier on camera and no prosecutor would have dared touch him.