Again, it's not just Steve that veto's Stark. Pretty much all of them except Reed did... which is understandable since AGAIN he's inverted. Reread Axis, then ask yourself how much you're going to trust an inverted hero on that boat.
Also, it's probably worth noting that it wasn't just Steve ticked off at Stark. Captain Universe seemed pretty peeved as well. That's basically the universe itself taking Steve's side and lashing out at Stark. So there may be more to it than Steve's injured pride (though that likely is a part of it as well).
Inverted or not, trust worthy or not, you'd rather have stark on the boat building things than not.
But that's sort of my point. Characters being written to fit a narrative rather than in any sort of consistent or sensible way. These characters have known stark for decades. He has personally saved their lives dozens of times- and they have saved his. They know him. They know what he is capable of. Even if they are distrustful of the current amoral Tony stark the idea that they wouldn't want him on a mission to rebuild humanity is laughable.
He attempted to murder Steve in axis. Having a guy that could potentially try and murder you is a valid reason to leave him off the boat. Which isn't to say there aren't valid reason to include him as well, but ultimately everyone weight the risk versus reward and made the call they did.
Ultimate Reed should replace 616 Reed who has been shown to be completely useless. 616 Reed refuses to both learn magic and up his scientific game to reduce magic to science. At least Ultimate Reed has the will to reduce magic to science enough to kill the Ultimate Asgardians.
In Hickman's Fantastic Four storyline one version of future Franklin seemed to do just fine after being orphaned, so it's not like 616 Reed is essential. I still think 616 Reed, because he was destined to be killed off by the Mad Celestials, before future Franklin altered the timeline, is now locked into now being ineffective. He's dead weight.