It's not mentioned in passing, it's stated in the beginning that the Straw Hats would bring the Dawn of the World and then followed up by her curiosity on why this would be Pedro's last venture. He then told her before he ordered her to help the ship set sail is that she
will understand their importance and their voyage. She was already helping the Straw Hats for seemingly no reason outside a desire for adventure, but Pedro meant a deeper reason for him lending his aid. Meaning that the knowledge will eventually come to her when she realizes what the Straw Hats are striving for that both Kozuki Clan and Mink Tribe have awaited "for centuries." Pedro died after laying down the foundation, it was up to the rest of the Straw Hats to carry that out since he knew ever since Luffy was shown the Road Poneglyph he would be the one to do so. If it didn't involve Carrot, I once again ask you: what was the point in her being told this if it didn't matter to her? Pedro could have had an internal monologue and it would have had the same effect.
That's why I say the inherited will of Pedro was to see the Dawn to its fruition. By the time he fought Baron Tamago, he already knew the Straw Hats would be the ones who will do it. All he did during the time as the Nox Pirate captain he just laid the foundation so that their journey will become easier when the time comes...which is about now.
The journey he wanted as a kid, but needed a moment to "shine." Which he imparts onto Carrot.
It doesn't ring the same where there's no payoff on Carrot's end to note this.
--- Update From New Post Merge ---
Except you knew the purpose behind Kinemon/Momo, and Law. One was freeing the country from the tyrannical reign of Orochi and Kaido as well as completing Oden's will of opening Wano's border. The other was the person who pretty much set in motion all the things that have transpired to now: -> destroying the SAD factory's production of DF and kidnapping the creator -> break a hold of an underground broker supplying said DF -> igniting the wrath of Kaido. All of Law's actions have led to the eventual confrontation of Kaido.
So no, comparing these 3 to Carrot are not even close. Since, once again, you don't why she's been as prevalent with the Straw Hats till now. You could have had Pedro and Pekoms do
any of the things you guys claim she was there to do. Both could display Sulong: Pedro was solely allowed by Nekomamushi to accompany the Sanji Retrieval Team assumedly because he was the
only one who could calm down Pekoms if the need arises. This was confirmed by Pekoms later in the arc. Pekoms had a richer history with Pedro considering he was a member of the original Nox Pirates crew before defecting. In fact, Pekoms even knows of the circumstances of Pedro's lost lifespan, something Carrot herself didn't know. So ignoring all that and comparing her to them is rather disingenuous considering no one knows why she's constantly helped out the crew for this long with seemingly no narrative purpose provided.