Get the SysTest binaries from Google Drive:
https://github.com/keirf/Amiga-StuffEither use the ADF to write a bootable disk, or run the executable from WB.
Use the real time clock test to see what is read from the M6242 every second. If you use the reset time/date function you should see it counting up normally from midnight. If that's not working correctly, then there's probably an issue with one of the clock or data lines to the M6242.
Normally the Amiga reads the hardware RTC once at startup, which it uses to set the time of day counter, which is then incremented by a timer in one of the CIAs. When something does a system time request, then usually it'll read the time of day counter as opposed to the hardware RTC.
Not sure what happens behind the scenes with the WB1.3 clock prefs, but SysTest is a more reliable way of seeing what the hardware is doing.