The page in question doesn't prove anything either way for Carrot or Yamato. It's a panel about the ones fighting the Tobi Roppo. Chopper, Brook, and Robin don't appear, nor do King or Queen. It's really that simple.
As for the Perospero fight - the whole fight doesn't need to be happening at all. It's only happening because Oda initiated the plot thread. It's weird to think 'it's unimportant so Oda will off-panel it'. Yeah, okay, but then why did he start it? And do we really think the story of the Charlotte Family ends on Wano? It's backward logic - by virtue of the fact that Oda is even continuing the storyline at all, despite the fact that it seems ancillary to the battle against Kaido's crew, is in itself a bit telling. The fight is important, but we don't really know how / why it's important yet.
My guess, and it is a guess, is that Big Mom's story doesn't end on Wano. I really struggle to imagine the alliance beating both Big Mom and Kaido. They're struggling against Kaido alone and it will probably take more help (waiting on Moria and 3 others) to defeat just Kaido. I really can't envision a scenario where they defeat both Big Mom and Kaido in this battle. The most important members of Big Mom's family aren't on Wano - and even if there were, the Straw Hats are already occupied with the Tobi Roppo. It's unclear who's even going to defeat King and Queen much less characters like Oven and Smoothie should they arrive. Katakuri, Pudding, Lola, Chiffon, and Brulee are all missing and Big Mom still has narrative ties to Elbaf. Capone's cover story arc reunited Lola and Chiffon and given Oda's track record with cover story arcs, I can only imagine that storyline will fold back into the main narrative in the future.
There's no one left to help Carrot and Wanda fight Perospero. So what this means is either Carrot defeats a 700 million bounty pirate, which is itself a huge feat, or, what I'm thinking now is more likely, Perospero wins. And then what? Carrot doesn't avenge Pedro… and her story just ends? Or does she meet Perospero again someday, say on Elbaf? It's interesting Carrot is present when Chiffon tells Nami about Lola's engagement to Loki and a thought bubble shows her considering the existence of giants with amazement.
--- Update From New Post Merge ---
@Syphin:
With the page expressing the narrative of the New Generation facing off against the Officers of the Beast Pirates (I assume the CP0 referred primarily to the Tobiroppo), it seems apparent that the characters focused on in the panels will be the ones engaging against the Tobiroppo, at least for the moment.
We know Franky is taking on Sasaki.
Sanji is attempting to escape Black Maria while having his duties conflict against his principles.
Jinbe is battling Who's Who.
The only two Tobiroppo members left in flux at the moment is Ulti and Page One. Based on the panels within the page, I suspect Ulti and Page One split off to go after different threats.
I expect Page One to go after Nami, Usopp and Tama. (Tama may eventually convert Page One with her abilities).
And Ulti will once again challenge Yamato who is protecting Momonosuke and Shinobu. Considering the history Ulti and Yamato have and the narrative in which their character panels are presented, I expect the two to clash. Ulti was taken out by Yamato earlier in the arc and has not repaid the debt back to Yamato. It seems inevitable the two will fight once again (it could just be me but Ulti is looking up while Yamato is looking down).
There is still the All Stars to deal with. With Marco taking on both King and Queen, Robin and Brook may encounter Jack. Eventually, I suspect Zoro will leave Kaido to Luffy and switch his attention to either King or Queen. After Chopper prepares the antidote, he may end up engaging in a team battle against Queen if Zoro ends up facing King.
Tama and Yamato are very much directly involved in dismantling the power that has reigned over Wano Kuni for the past 20+ years.
I don't really think Zoro will leave Luffy's side, nor do I think he could handle King or Queen. Combined Luffy, Zoro, Kid, Killer, and Law are struggling to damage even Kaido without even attempting to attack Big Mom. If anything, they need more help. I think there's very little chance Luffy defeats Kaido solo in this story arc.