Yeah, they only work for Fisk. This is kind of a carry over from Rosenberg's Punisher series where Fisk was working with Baron Zemo and relaunched the Thunderbolts as Fisk's personal superhero team(that lineup was Zemo, Moonstone, Fixer, Ghost, Radioactive Man, and Jigsaw. They didnt really do anything and the Punisher beats them all). This iteration is just Fisk's response team to Knull's takeover of the world, and he makes it extremely clear they are just basically human shields for Star in this situation.
This run is only four issues and starts with the preface that all members of the team have died in a flash forward press release by Fisk, I assume some will survive but just be considered dead to the public, either as a reward by Fisk or just faking their death.