CRP can’t really confirm an infection. It’s a marker of inflammation. Inflammation often correlates with infection but not always. Can be lots of other things too, including a lot of the things that are false positives on these sepsis algos.
Ultimately doctors have to take the whole clinical picture into account instead of relying on a couple markers or vital signs.
You’re right that a big problem is the inability of our system to do testing at home, which might greatly expand the scope of the clinical picture. But it still wouldn’t be a cure-all