Cool stuff! Back in 1972, Signetics published a booklet on their then-new NE560 NE561 NE565 AND NE567. My memory is a bit dusty and I don't remember if the NE564 came later. I think there was also an NE566 which only had the VCO of the NE565 brought out to pins. It was a big brown book though not too thick with green print. Maybe you can find a scan. Mine is long gone.
This thread tripped a memory. The book had a table comparing the NE56x series features. Either the NE561 or the NE560 had a demodulated AM output that had been optimized for linearity. Now unobtainable...sigh. HOWEVER, it also indicated that the NE567 had both FM and AM but not optimized for linear demodulation. I tinkered with it a bit back then, and also played with a sample and hold capturing the tiny positive peak window for best S/N. The simple 567 as a tone decoder would lock onto the carrier rock solid and I used old style one shots with trimpots to set the capture window based on the PLL.
If you can find the old book, it will give you a lot of ideas that can be used with both the older PLLs and the newer ones. Anyway, the 567 can be a lot of fun in the world below 500 KHz.
73 de Jack W8ADQ