I know support people will tell you anything to get you off the phone, but saying ATV needs to be directly connected to the TV is ridiculous. It works absolutely fine through an HDMI switching receiver (I also have an Onkyo). On the very rare occasion I've had an Airplay issue (maybe twice ever), the thing that's fixed it was rebooting the network everything is connected to, i.e. wireless access point, router, switch if you use one, then finally ATV. Perhaps the high traffic level is too much for an always-on router if there's an underlying issue that increases over time, for example a memory leak in the router firmware. I haven't had to do this a long time now, but then I have a different router nowadays.
I have several devices connected via HDMI to the Onkyo AVR and from there to the TV via HDMI, and it all works perfectly. In fact, the only HDMI issues I've ever had were back in the day before getting the AVR, and directly connecting things to the TV. Back then, HDCP handshake errors were not uncommon. So for me, the AVR has fixed problems that direct connection could not seem to.
If this problem is repeatable with only this particular content, then it's not completely impossible it's a bug in Airplay either in ATV or in the sender. If it's repeatable with a variety of content, then it's more likely the network or HDMI cable, or I suppose there's an outside chance the receiver is faulty.