You have to compare to the current situation where we are overreliant on fever as a marker. If after an operation, your CRP levels shoot high unexpectedly, wouldn't you want to know and start investigating what's going on?
CRP normally does shoot up post operatively, and white cell count behaves similarly. Generally if there is a post op problem people know- maybe not a fever but people feel generally unwell. That is a good reason to investigate further. Sometimes for higher risk operations we do routine blood tests as an inpatient, but if people are will enough to go home, it’s reasonable to rely on people not feeling right.
I think what you are assuming is that early intervention in infection is good. Very likely to be true. And that we can reliably detect early infection. Much less likely to be true.
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.
You should be doing both. WCC and neutrophils can also be misleading, and tend to fluctuate. Procalcitonin can be helpful in some situations but isn’t used as often.