Identifying an unknown code "W62" SOT363 dual inverter
Based on the net connectivity of the unknown circuit element U50, I would strongly believe the component to be a dual inverter. Using an inverter as the active gain element in a Pierce oscillator, is very common for non-RF applications.

You can test the basic functionality of the component by over-driving pin 3 (based on your hand drawn schematic). Connecting pin 3 to ground, pin 4 should output high. The other gate in package should also perform its correct logical operation. 

Aside: The 2 load capacitors (on either side of the crystal) would most likely be in the range of 16 to 64 pF depending on the crystal used. Very unlikely to be 100 nF.