I think the World Government planned on waging war against each of the Yonko (or Sanko as this was before Shanks became an Emperor). Why? Because they were scared. Roger did the unthinkable. He sailed to Laugh Tale and learned the true history of the World. Something the Government feared more than anything else. But since Roger was too early and he had an incurable disease, he couldn’t do anything and the World Government dodged a bullet there.
But this still scared them, because there were others who are strong in their own right or were becoming strong and they needed a way to stop them. If not, they might end up doing the same thing Roger did. In particular they had to stop 3 people.
“Whitebeard” Edward Newgate
I think the World Government was interested in the Gomu Gomu no Mi because they thought the fruit would be a hard counter to Whitebeard’s Gura Gura no Mi. They might have thought the rubber fruit could absorb and/or nullify Whitebeard’s quakes if awakened. For those who don’t know, in Japan, Rubber is used in buildings to counteract the effects of earthquakes. Rubber is used a shock absorber to prevent massive property damage.
We know Whitebeard technically had the entire world at gun point since Sengoku admitted that Whitebeard could end the world at a whim.
I think the World Government knew this and had been planning countermeasures for a while.
One such countermeasures to the quake devil fruit is the rubber fruit. Which if theoretically awakened would allow for the effects of the rubber to be spread across let’s say, Mary Geoise and make the entire city immune to Whitebeard’s quakes.
“Big Mom” Charlotte Linlin
The World Government placed a 5 Billion Berries bounty on the Ope Ope no Mi. We think this was because of its relation to the National Treasure of Mary Geoise but I also think that Law’s devil fruit could be used to counter Big Mom’s Sor Soru no Mi, which is why he might be fighting Big Mom right now.
Law also has some sort of soul related powers with his personality transplant and all that. I don’t exactly know how it could counter Mama’s fruit, but I guess we’ll find out shortly. Maybe Law could swap the souls of the Homies so they might not be able to use their powers properly, severely nerfing her.
“King of the Beasts” Kaido