Had exactely the same problem since I got my IIfx 5 years ago... The first thing I made when I got it was recapping but the problem was still there!
This WE I went on the problem to see if there was any broken trace with the power circuit and RTC schematics... but everything was fine...
In purple, the traces I tested were all OK!
At last resort I decided to pull a RTC chip from an old mac Classic in order to replace the IIfx's one... and... it works, clock is counting!!! What a surprise since I had never seen a RTC fail before!
I still have the power off issue (I have to unplug the mac for shutdown), maybe I'll have to replace the Nand gates and the related transistors?