Yes, Doflamingo was by far the most influential and important Pirate in the New World, even more influential then any of the Yonko.
Doflamingo and his family were basically running the entire Underground of the New World which in a way also means that he was the most important person in the New World as well.
- Kaido’s entire dream of creating a zoan army shattered after Doffy was imprisoned.
- Doffy supplied many islands with weapons to wage war and after he was gone they had no means to continue their fight anymore.
- His authority and political power even outclasses that of the other Emperors since he blackmailed the Government and in return received authority over CP0 and the Marines.
- His defeat started the battle for the Throne.
In his speech he even states that all of the big names were “fed” by him and now that he’s not around anymore everybody panics. He controlled the big timers of the New World.
During the Wano arc we even get to know that the World Government received their Seastone thanks to Doflamingo’s connections to Kaido. Now CP0 has to step into a disadvantageous position and negotiate with the Shogun personally taking huge losses.
Also makes sense now why they immediately rushed to Dressrosa to clear his mess up since they want Doffy at their side at all costs.
There are also many arguments showcasing that Buggy’s business can’t fill the void that Doffy left. Basically everybody’s business (apart from Buggy) got screwed because Doflamingo was taken down.
*Theory by nukiri6