I don't have time to answer this right now in any detail, but the reason is basically that the psychological development of newborn infants DEPENDS on the act of nursing, albeit for only a relatively short while.
That's entirely not true. Bottle fed babies are not psychologically effect by not nursing. If you want to just say as a rule that the ring gates won't work, say that and that will end it. But I'm not swayed by your reasoning.
After that point, things can become much easier on that front - but I'm telling you that there are simpler solutions than the ones you're trying to create. A nursemaid is the SIMPLEST option, and though that nursemaid might die, there has to be a point at which you judge the life of a child *more important* than the life of someone else. If Jenny the peasant mom dies to defend the life of the secret princess Arendorr, you'll make sure her child is cared for (or maybe she already lost it) but you can't imagine that the sacrifice wasn't (or wouldn't have been) worth it.
But why endanger someone's life when it's not necessary. Not only does taking a peasant along with us endanger the peasant, it endangers us. An adventuring party is only as strong as its weakest link. Looking after a nursemaid endangers the circle even if we bring in another warrior for that purpose. An infant can easily obtain full cover if you hold her in a shield arm. A nursemaid is much more difficult to protect.
Plus, YOU keep saying that it's important for a child to have the same milk for the first few weeks of life. What happens if Jenny the nursemaid dies in week one? Then we're up the creek without a paddle. If she stays in Hyrwl and nurses through the ring gate, then she's one less thing we have to worry about.
This child is the last hope if the current line is corrupted - a HUGE burden will be placed on her shoulders from a very young age. Turning to magical maintenance of the child is not the best way to do make sure she's ready for that kind of burden.
What's wrong with using magical maintenance in a world that relies on magic? When we discussed the Transfer Pregnancy spells, you said that it would be a standard for nobles to never have to give birth. And children of nobles are essentially handed off to servants for care and upbringing. I would think that bring a child up with her parents would be less damaging even if you feed her through aid of magic. How is her nursing milk going to determine how prepared she is for burden? It's her upbringing that will determine that. Even if she's nursed through artificial means, it doesn't mean she won't be loved and cared for.
LASTLY, take it as a DM hint, using a boobie-gate is not a good idea. There are a uncountable ways that the cruel universe could turn that against you, and the child.
I've thought of and accounted for several of these. Fortunately, the circle already has active means to counteract them.
1) Someone could come in disguised as the nurse. Well we have a circlemember with a gem of trueseeing. It'd be easy enough to inspect the nursemaid before each feeding.
One of the benefits I see in this is that if the nursemaid turns out to be evil, feeding through the ring gate, one of the circle members will always be holding the child. If a nursemaid traveling with us was holding the baby, she'd have an advantage if she tried to do anything to her.
2) The nursemaid could poison the teet. Well she could do this anyway, even if she were traveling with us. Detect Poison and Status aren't that difficult to cast.