Like at North Anna.
If you presume he won't retreat (Which even Longstreet realized - he said something like "Grant will fight us until he wins") the worst case scenario for Grant pretty much happened anyway given how the ANV was able to get to Spotsylvania. In fact, it's possible that the ANV doesn't feel the urgency and the AOTP reaches it first in TTL, as it was a close race.
If thigns go pretty much as OTL till North Anna, I think Longstreet might be able to deal a harder strike to the AOTP, as Lee was hoping Hill would do, but Grant's still not going to retreat. Let's say he loses more men - he's going to move to the better ground, and then I can envision a repeat of the situation at Gettysburg, where Lee keeps ordering attacks even as Longstreet says it's not going to work.Longstreet might not be as insistent - it'll be a much more dangerous situation for the ANV than it was when the Union was on the beetter terrain. But, as you say, Grant will find the better position and continue making thsoe flanking attacks, and it'll cost them int he long run. Instead of Lee seeming to be spent and Grant miscalculating at Cold Harbor, you might get a reverse of that, perhaps.
Either way, I doubt Grant attacks at Cold Harbor the way he did because he'll realized Lee's army isn't beaten. He might just go arounda nd lay siege right aay.