I think 74HC595 is the wrong chip for this application. It has to do with the minimum input high voltage for 74HC595 pins.
The minimum input voltage is calculated with 0.7 * Vcc. With 5V supply, the input voltage should be > 3.5V but the max output from a Teensy 3.6 is 3.3V.
74HCT595 (Nexperia/ Diodes inc.) or 74AHCT595 (Texas Instruments) is the recommended type. The minimum input high voltage for these is 2.0V.
Thanks!
I'm going to read the datasheet.