Yamao's forced because she's a character introduced out of nowhere in the final stages of a 6 year long saga that's demanding to join the crew, despite not knowing any of them.
Brook and Franky got arcs full of development with the straw hats and had tons of interactions with them that it only felt natural for them to join afterwards. Yamao has all of that development and shared panel time, but instead of it being shared with the straw hats, it's shared with Momo.
If every random Tom, Dick and Harry that was interested in the straw hats got to join the crew then they would've picked up their lot during their return to Sabaody.
You could say the same about Carrot. She's been introduced since more than 2 and a half arcs ago, yet she has yet to interact with every member of the straw hats.
Yamato despite never meeting in person every member of the straw hats yet, can already recognize them from a glance and has taken a special interest in them, like when she recognized Franky. So saying that Yamato doesn't know any of the straw hats is false. Between who knows more about the Straw hat crew, Yamato is definitely ahead of Carrot in this regard. And like
@Garp the Fist already said, Yamato also personally knows Ace, she understands Luffy's pain and loss just like every other straw hat. Carrot doesn't.
You call Yamato forced because she's introduced too late into the arc, when Carrot was forced onto the Sunny literally one chapter AFTER Zou when she snuck on the ship.
You cite the lack of interactions of Yamato with the straw hats on Wano, but that's happening right now. It's going to increase more as the chapters go by, and will be completed by the end of the arc. In comparison, Carrot is the one actually lacking here for a potential Straw Hat- despite 2 and a half arcs, she still hasn't properly interacted with every straw hat yet. It's the longest record that Carrot has managed to break after Sanji first properly interacting with Zoro in Arlong Park when they first met in the Baratie Arc. She clashed with Zoro and they never talked again. Carrot only got to interact with just half of the Straw Hats in the WCI, then come Wano and that interaction was never finished for every member of the crew. In the current Wano Arc, Carrot has been swept back into the other Minks where she belonged, got off paneled by Perospero, and we never heard from her again. Carrot's interactions with the Straw Hats is nothing to celebrate about if we're being honest.
Comparing Brook and Franky's development vs Carrot is also a bad comparison imo because both of them got long and dedicated backstories and flashbacks that were not only long and thorough, but also are heavily related to their respective arcs. Brook lost his shadow to Moria many years ago, and only got it back after Luffy and the Straw Hats helped Brook take down Moria, who took their shadows as well. Brook was also the lost and last surviving member of the Rumbar Pirates, who just like Luffy and the Straw Hats share connections with Laboon. Franky lost Tom and was wanted by the CP9 and the WG many years well before the Straw Hats reached Water Seven, he also shares a connection with Robin as the 2 crucial people to revive the Ancient Weapon Pluton. Franky and Robin met in the Enies Lobby Arc, and they left together with the Straw Hat Crew as well. Yamato's history with Ace also makes her encounter with Luffy many years in the making. Yamato's struggle to be free from the literal shackles of her father Kaido coincides Luffy's goal to liberate Wano from the tyrannical rule and oppression of Kaido. The only backstory or flashback we got for Carrot was Pedro giving her electro claws because she wasn't good at using a sword, lasting just a few panels. That little flashback had nothing to do with the history of Zou, the Road Poneglyphs nor "The Dawn" (which even now hasn't even been properly explained what it actually even is).
Yamato is not just another Tom, Dick or Harry. That description is more fitting for Carrot than Yamato if we're being serious here.