The problem with hanging your hat on the lookout argument is that it’s such an unimportant position, literally anyone with good eyesight can fill it. Furthermore, we already have a sniper and a navigator who can both see targets/weather patterns from miles away...since we have no feats whatsoever to judge her by (the example you’ve been using is not what I would call an accomplishment), are you really comfortable saying that Carrot’s vision is superior to Usopp’s or Nami’s?
You're changing the topic entirely,i was just explaining why the page that thedarkdragon11 has shown wasn't a good argument for "she's a bad lookout!11!!!" But i'm gonna try and explain your questions. Take your favourite drink and enjoy.
It's kinda obvious that any position post-ts would be "unimportant" because the crew got to New World without said positions.
You said that it's "Not an accomplishment" ,but for me if you have no Observation Haki,yet you can detect things faster than Observation Haki users makes it look like an accomplishment(there was another scene-when Reiju first appeared,Carrot once again was the fastest one to notice her even though we had Pedro and Pekoms on board - again, both Observation Haki users).
As for her usefulness as a Lookout. In the SBS(Volume 74) it was stated that ALL Straw Hats rotate on the position in question resulting in lack of sleep. Carrot has been introduced in Zou as a Ruler's Aide,which got explained to be a position in which you live at irregular times to serve both Inu and Neko. That solves this problem,the dedicated lookout would take that weight off everyone's shoulders.
Correct me if i'm wrong,but Nami can only read weather patterns and stuff but when it comes to noticing enemies she isn't good at all,after all if she was that good then we wouldn't have countless panels of Carrot being the one telling Nami what's coming,not the other way around,and i already provided evidence above. And Usopp? Yea,he's a godly sniper,but the role of a Lookout isn't for him. Remember what happened after Punk Hazard? He was scared of Doflamingo coming after them,what does he do? Well,not go to the lookout post to keep watch if he actually is coming... And i know you'll bring up what he did in Chapter 978,and honestly,i can't defend Carrot there. She slacked off in that instance. But honestly speaking,if we are told that all SH's rotate on the position it's rather clear that Usopp is not better than others at it. Also i'm sure that Carrot already makes up for more than half of the panel time where someone was on the Sunny's Lookout post,only Carrot made that spot relevant,not even the thought of Doflamingo coming after them made the position have any relevancy,only when Carrot was there for an arc it got some shine...
Furthermore,i remember a scene(i think it was pre-Mock Town?) where Robin was on the lookout post and Luffy was rather disappointed in her showing and explained what the true lookout should be - cheerful and being excited at a sight of an island. And that's basically how Carrot acts,even early WCI confirms this,where Pedro wasn't particularly interested in what he saw,but the moment Carrot got there we got info on how the Tottoland seas work,all thanks to her nature.
And here you have it,my explanation on why Carrot is the perfect option for the position.
Of course it isn't anything groundbreaking and you may view it as "unimportant",but as i said,it would be naive to think anything could be introduced so late when they got so far already...
If you made it this far,good job. Though I doubt i convinced you. I think that arguing about this is pointless, and i'm sure you smart guys have different problems about Carrot that i even could agree with than this.For me in this category(what role would she have) Carrot clearly is ahead from other candidates.Of course she struggles in different areas,but that's a different topic.