Why is an approach that relies on both LOC and PAPI not considered a precision approach?
I think you're overlooking the primary reason pilots fly instrument approaches: the conditions do not permit flight by visual reference.

The fact that a PAPI provides vertical guidance is more or less useless if you can't see it. Most ILS approaches involve a descent of at least 1500' from glideslope intercept. 

If you break out from the clouds at ILS minimums, you can be as low as 200' above the runway surface; at 3°, that's around 3/4 miles from the touchdown zone and the PAPI. It's highly unlikely that the PAPI would be visible prior to that point, meaning the approach would be made with purely lateral guidance.