I'm not saying that Zeus is going to be a good parent; his aloofness and selfishness are very evident, and even if he cares about the Bearer, love itself is a double edge sword. Just that in some ways he'll try to show his care as time passes.
The rumor about the Bearer being his favorite honestly comes off as foreshadowing for what experiences the Bearer may have with Zeus and Hera. In the original opening, it was hinted that as he held the baby Bearer, he did feel some hint of affection for them (regardless if the child in his arms is his own or not). And it's pretty obvious that Hera isn't too fond of the Bearer herself. So I can see a non-Zeus Bearer getting a lot of more grief from her due to her sense of wounded pride due to the Bearer's lack of 'blood ties' with Zeus yet Zeus STILL favors them (I really do feel like she'd actually prefer that Bearer WAS related to Zeus in some way). And I can see Zeus trying to create some type of bond with the Bearer (no matter the background) due to either some genuine affection for the Bearer or his (possibly) genuine affections for their mother Priscilla.
That said, I found his reaction to the Bearer wanting to attempt to give him a chance somewhat interesting, which kind of makes me think that. He seemed genuinely relieved about the choice and seems about to say something before just shaking his head. Like even he knew that it is a bit much to have expected the Bearer to be willing in any way to have a familial bond with him (not that he would admit to it in any way). My impressions of the scene, of course.
Pan is a bit indifferent to bonding with Daddy Dearest Zeus. Not because he hates the guy, but because he already had parental figures in his life when he needed it the most and he's at a point where his independent nature is at its strongest, so he's not look for anyone's approval (he's just going to do what he wants). That said, Pan is willing to give a chance to his father, though he won't be the type to actively seek out the God, and will simply interact when the God seems willing to initiate interactions. Nyx (his daughter) is just as self-assured as Pan and while she's more likely to try to start the interactions between her and her father due to her more outspoken and friendly nature, it's no skin off her nose if the guy can't do it.
Ironically (and possibly much to Hades' dismay), I feel like Damon would actually be more likely to create a sense of kinship with Zeus first before rekindling a relationship with Hades. He's not really looking for a father figure; just someone to acknowledge his growing strength. And as selfish and aloof as Zeus is, Damon would probably find it more honest than his Uncle's cryptic words and habit for dancing oddly around a issue. There's also the fact that he's got a more neutral solid ground with Zeus and the other gods (though with understandable wariness involved), while Hades is unfortunately on very shaky grounds with Damon after the reveal that he is Damon's father.