Reconciliation: the only benefit to the RF would be for Charles - there are plenty of British nationals who believe that they would never let anything stand in the way of a relationship with their child/grandchild and judge Charles for that. But the risks for Charles are enormous (Harry's proven lack of discretion). It's only been weeks since Harry announced a government conspiracy to make him 'less safe' and the King is apparently wilfully either participating or not getting out the way (endorsing it basically).
Harry is an idiot and I think he wants a reconciliation and he's still angry and he doesn't realise that those are contradictory positions. I think he has a fantasy of the RF apologising to him and then everything would be hunky dory (he's forgetting that the last thing M wants is reconciliation). But his anger means that he's very inconsistent. I could definitely see that appearing with senior RF members at Invictus would benefit his profile and garner a ton of attention, much of it positive for H and the charity. I agree with a PP, Sophie, Edward and/or Anne will attend. I think Beatrice and Eugenie might well attend too. I wonder if Charles, William and/or Kate might meet with athletes prior to the competition (without H).
Another big 'party' that would like reconciliation: the British media. H & M have generated far more headlines, access to so-called 'private lives' and juicy stories than they would have done if H & M simply kept their heads down and got on with the job quietly. Regardless of how critical the headlines are, any media that benefits from clicks, etc, has got to absolutely love the gift that keeps on giving. Reconciliation would generate a load more coverage and comment in a way that 'grey rock' does not and it'll be interesting to see if any of the British media attempt to put pressure on the RF to bring about a reconciliation of some kind.
William and Kate: I don't think they've got anything to gain. The biggest benefit to having H & M back would be if they could be full-time working royals because it would give George, Charlotte and Louis longer to live privately before they're needed for engagements. There's zero indication that H & M could ever be part- or full-time working royals again (and I don't think M would want to anyway) so therefore W & K can't have the biggest benefit to H & M's existence (in a parallel universe where they are team players). So, no, I don't see any way forward for the two brothers to be close in the future, or even really on much more than distant speaking terms. My guess is that they send presents to the kids on their birthdays/Christmas (and I think that probably happens in both directions) but that's it.