I read that your Onkyo doesn't support HDR10 or Dolby Vision, so depending on the capabilities of your TV, putting the Onkyo 656 in between may pose a bottleneck for advanced color features if you connect the ATV4K to it. Conversly, the Onkyo supposedly supports Dobly Atmos, but due to the high bandwidth way that ATV4K supports Atmos, your TV may pose a bottleneck for advanced audio features if you connect the ATV4K to it :-). I believe I would opt to connect to the TV if it's capable of HDR10 or Dolby Vision. OTOH, the Onkyo should support passing basic 4K HDR@60, so...
Have you followed Onkyo's video troubleshooting steps for the NR-656? I see that it specifies using HDMI ports IN1-IN3 for devices that support HDCP 2.2 (your ATV4K is such a device). AVRs and TVs often reserve advanced capabilities for specific HDMI ports.
To build on the general advice offered by @Tim_4_All, I always suggest digging into the settings menus of both your AVR and TV to turn on or enable all of the advanced auto-configure (HDMI CEC-related / Video HDR-related / Audio Dolby-related) functions available. These settings may be universal or port-specific (and may be spread across multiple sections of convoluted settings menu structures :-).
Next, reset your ATV4K's Video Settings while directly connected to the TV, back out of settings and unplug the ATV4K, then connect it to an IN1-IN3 HDMI port on the Onkyo and plug it in. Go to Settings > Video and Audio > VIDEO: Enable HDR and choose the best/ recommended chroma etc.