If it's of any comfort, I have two 8595E analyzers, and neither of them show anything during FM COIL calibration. I also don't see a sloping trace when examining the FM COIL DRIVE as described in the Assembly-Level Repair Guide under "Verify the FM coil driver". I do see a negative sloping trace when examining the FM SPAN.
Both units work fine and self-cal successfully.
The FM coil is used to sweep the LO for spans of 10MHz or less. You could take a look at the FM coil voltage and make sure you see the ramp. Probe either side of R902, which is one side of the FM coil (FM+), at the top of the A7 analog control board. You should see the FM coil slope and it should change with the span. R902 location and scope capture below.
The FM+ signal is what should be monitored during CAL and during the FM COIL DRIVE diag display, but for some reason it is not. It does not appear after the analog test MUX (ANA_TEST), which can be monitored via TP2.
My guess is it's probably a firmware problem that the test MUX is not getting set up properly for the FM+ signal (FM COIL DRIVE, aka FCDRV). If you want to look into it further, the MUX is detailed in sub-section "B" in the CLIP #5963-2951, page 280.
You could also ask about what you're seeing in the HP-Agilent-Keysight-equipment group on groups.io.