When we first meet Captain Lorca, he says quite plainly that he has discretion, and that Star Fleet have given him a mandate to do just about anything. While we have yet to receive any independent confirmation of his claim, it's clear that everyone else on his crew, including his XO, his now late Security Officer Landry, and Lt. Stamets, who seems to be Chief Engineer (but wears science-silver, so maybe he's actually Science Officer?) believes it. Landry was a fan; Stamets isn't; Saru appears to be having misgivings.
Burnham herself seems to agree in the general sense with Lorca's basic premise that Star Fleet must win the war. However, both her reaction toward the end of "The Butcher's Knife..." and clips from the preview for "Choose Your Pain", make it clear she's deeply uncomfortable with with the specific solution to making the drive work.
It's actually too early to know what Star Fleet thinks, beyond, "Yay! Someone saved Corvan II". The research is double-plus-top-secret. USS Glenn was operating as much as a black-ops ship as Discovery. Admiral Cornwell herself may yet have no idea what the details of the drive's function are. Indeed, we can probably assume she doesn't--otherwise, Burnham's deductions would not have been necessary.
If nothing else, I think we now have the answer, surprisingly early, as to why the Spore Drive did not catch on and replace Warp Drive. If there's no way to artificially reproduce the results obtained by slotting a mega-tardigrade into the system, and there's no way to obtain its consent or mitigate the negative effects upon it, there's no way a post-war Star Fleet would be willing to continue to use the system.