Although the Mac II uses a 661-0375 PSU and the IIfx a 661-0542, it's possible that this is more relevant to the shapes of the cases than the electrical design. The difficulty probably lies in the 5V TRKL supply from the PSU, which is present while ever the Mac is attached to the mains supply, and which is necessary for startup.
Only one of the batteries is concerned with PRAM settings, but both should be 3.3V or more in circuit, and certainly replaced before they reach 2.8V. Shutdown depends on the same TRKL supply's having already fallen to less than 1V when the command is given. You don't usually see the message 'You may now ...' because it is either circumvented or too fleeting when correct shutdown occurs.
One of my IIci Macs starts correctly with soft power, but declines to shut down in the same manner as your Mac does. Apple's Service Source says to change the power supply. 15-odd years later you live with the fact, or adapt, or try the PSU repair here.
The PSUs of the IIcx, IIci and Q700 are identical (661-0467), although the Astec units fail in the TRKL circuit more often than the Delta PSUs. Failed units can be started as you have done, or by 'bumping' (a quick off and on again with the mains supply switch), or perhaps (if you can verify that the same sawtooth oscillator circuit as in the IIcx &c. PSUs is the cause) by the repair cited.
de