Does a Prince in Vampire: The Masquerade get to automatically make their own childer into neonates from fledgling status without having to go through a presentation process to someone else? I would say probably yes, but it is a bit unfair if everyone else has to put their fledglings at risk of being torn apart for failing this rite of passage.
The third and fourth traditions are absolutely bent in favor of Princes. That's the way that the founders of the Camarilla wrote them. It is certainly good practice to foster your childer before releasing them (you still have to release them), and it can be a social coup to have a coming out affair where Princes from neighboring domains come and "see" your childe without actually needing to offer acknowledgement. But the Second Tradition means that the word of the Prince is law in his or her city.
Is it unfair? Absolutely. It's one of the things the Anarchs hated, which led to the Revolt.
I would posit that the questions that should be asked are as follows, based on your vision of this city in your V:tM game:
Who is the authority above the Prince?
Typically, the answer to this would be no one. The prince rarely has any approving authority above himself in his domain, hence his title.
Who would the Prince present their childer to?
This is, historically and traditionally, something that occurs for the prince themselves. Even elders are required to present themselves to the prince. That said, more than the Prince are typically at Elysium when a childer is presented to the Prince as a neonate.
What will happen to the childer if not presented to other kindred of the area?
Most unannounced neonates would be noticed within the city as strangers to it relatively quickly. The moment this happens, the city's kindred would bring this neonate to the Prince anyway. It is actually in the Prince's best interests to present their kindred to avoid any accidents befalling their recently created childer.
The title of Prince conveys an air of authority of the region that is supposed to be the ultimate authority of that region. They determine who is allowed to make childer and who isn't. Yes, they could technically make any and all the childer they desire, but if they go too far overboard, there is a probability of upsetting the rest of the kindred in the region and inciting a coup, ultimately ending in a new Prince.
A Prince is likely to present their childer to any secondary ruling council of the region they control as a show of respect to that council and to ensure that everyone could see their new childer at Elysium. This would stop the Elders from seeing this new kindred in town and abducting them to present to the Prince.
The laws are up to the Prince his/herself, but they do need to consider the other kindred of the region and the status/likely survival of the childer that they create if they do not show them.
Regarding permission to sire, on page 36 of the Revised Edition core rulebook, under advantages of Princedom:
Right to Progeny -- Only the prince may freely create progeny. Other vampires who wish to sire must first obtain his permission or risk the destruction of themselves and their new childer. [...] conversely, he [the prince] may sire as he chooses, [...]
Emphases mine. If you are referring specifically to the ritual/ceremony of presenting a childe to the prince as the childe's formal entrance into Kindred society, on page 39 of the Revised Edition core rulebook:
It is a popular Camarilla conceit that a sire recite the Traditions to his childe before that childe is recognized as a neonate. Some princes stage grand spectacles to honor new childer's transition from fledgling to neonate, while others need not even witness the release, trusting the sire with proper execution.
Emphasis mine. The presentation ceremony, while sometimes a big deal, is clearly not an ironclad requirement in the Camarilla. If the Prince cares, the ceremony might matter for other sires (though it sounds hard to fail badly enough for childer to die over it). It's hard to see how this could go poorly for the Prince's own childer.