I have been in a similar situation where after several days of illness my condition didn't appear bad enough and the doctor sent me home. This was some 15 years ago and they didn't have quick tests back then, but luckily they did prescribe a lab CRP test and called me back in for treatment immediately after they got the lab result later that day.
A lot depends on your index of suspicion, how well the person is, what the risks are of getting it wrong (in both directions), what safety nets you have available to you (i.e. are they home with someone sensible who can keep an eye on them).
In summary - there is no perfect test or perfect heuristic to apply to everyone.
She already had a lot of burst veins from the hospital, so we decided to just do what her surgeon said, basically take amoxicillin for another 7 days and just about it.
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
The biggest issue is the costs of false positives, which with any early intervention protocol are likely to be high and, questionably, on balance a net negative.