There could be a role for Harry, if he worked with the Invictus Foundation and became some sort of asset for the Paralympics (in which there are always veterans competing at the moment).
Drawbacks: he doesn't seem to have any useful links to veterans' disability sport in US (the Warrior Games might have inspired him, but they don't seem to have any working relationship) and I can't actually think of any role he could usefully have.
I mean, I suppose he might hand out a medal, but that's usually done by people far more senior/experienced in international sport; (and that's the sort of thing that if you blink, you miss it - the Princess Royal, who has been involved in IOC for ages was handing out medals in Paris and it went largely unnoticed. IOC is of course likely to ask her to do that sort of role in 2028. But as she is involved with the main Games, that may leave the Paralympics as a possibility - the Duchess of Edinburgh is the British patron, but AFAIK she has no IOC role (and we have yet to see whether that's senior enough to do things like hand out medals in Paris, but I expect she'll be there supporting Team GB)
So my guess is that if Harry wants an Olympic role (and it's only speculation that he does), he's going to have to first find a role with IOC. And IG might be the only thing he does that is close to having the right credentials. What I'm less certain about is how much work Harry has put in to build the right kind of reputation to be considered for a role.