Why isn't an LPV SBAS approach considered a precision approach according to ICAO?
Precision Approaches (PA), as you mention, have specific performance requirements, one of which is how "good" the vertical position measurement must be during an approach. In the case of a PA, ground systems either directly measure the aircraft's vertical position on the glide path (Precision Approach Radar in the case of parallel approaches), or provide a means to follow a precise vertical glide slope via a gauge in the cockpit.

For Approaches with Vertical Guidance (APV), there is vertical guidance, but the aircraft's vertical position is measured onboard the aircraft by either a GPS position source or a barometric sensor. Either of those sensors are subject to measurement errors:


Barometric Measurement Errors: ICAO Annex 10 section 4.4.2.4 defines the error model that is used for barometric altitude reporting systems. The 95% bound of the worst case is +/-282ft.
Barometric Setting Errors: Additional errors are introduced depending on what barometric MSL pressure is selected on the altimeter and how that compares to current (actual) atmospheric pressures.
GPS Measurement Errors: While GPS position sensors provide very good horizontal position, GPS vertical position is generally poor. According to the FAA, the 95% vertical error of GPS systems used in civilian contexts is on the order of 300ft and according to this Advisory Circular (section 5-7), geometric/GPS altitude does not meet requirements for use in air traffic control internationally.


Combining these errors with flight technical errors while flying the approach, the total error you can expect can add up quickly. Compare that to the fact that in the vertical guidance of an ILS improves the closer you get to the ground, you can quickly see why the alternatives of GPS or altimetry don't quite cut it...

So specifically to your question: For an approach to be a precision approach, you need precise vertical position measurements, which an SBAS cannot provide.