Florencia's experiment was to save her brother and her son. Roswell was terminal with no prospects outside obsessing over a rat that had seen how things play out, and Oswin had given up hope living after his experiment and had to watch his son call someone else dad.
The reason Roswell didn't call it off before is because not going to the mansion doesn't solve the problem. He's tried, it's why we see glimpses of a timeline where Dave is a sole survivor, so that was a no-go. I think an important thing to remember is that Roswell's goal was to keep Dave alive, Thanatos's goal was to keep Roswell alive as much as he could, even in the face of Roswell's wants, and then Florencia just wanted her family fixed.
So why the medals? Because they're impossible to get in one run. They could've been anything at all, just a set of objects that she could've hidden in the same way. It would've required Dave to experience/remember the fact he's looped even with Thanatos as a witness.
The normal question now is usually "So does the medicine work or doesn't it?" the answer is "We don't know." It's a quantum mechanics thing referred to as a wave function collapse, though most people recognize the concept as Schrodinger's cat. All the possibilities of how this medicine works, if it works, is now consolidated into one outcome: which we know due to outside sources that yes, it will work.
As far as providing this explicitly in game, it becomes hard to treat readers with respect for their intelligence with things like this when the alternative is to just open a Wikipedia page on the subject and deliver it up-front. I could have Florencia or someone monologue for a while to get it all down, but at what point does it just become a clunky mess when in-universe Dave isn't going to understand anyway.