It's doing that because when the TV turns on the receiver (I'm assuming you're using the TV remote to turn on the receiver?), the receiver is switching to the TV scene which takes its audio from AV4. Try changing the TV scene to HDMI1. Read the manual to see how it's done.
Also, does Grandma need the receiver to be on? I ask because when guests/old people are using my system, I hide the receiver remote and use HDMI standby pass-through* set to pass the cable box. That way, it's like the receiver isn't even there.
* What model Yamaha is yours? Only the 473 and up have HDMI standby pass-through.
HDMI control only works with devices that support HDMI CEC control protocol. Sounds like OP's DTV box doesn't support that. You don't need to assign audio inputs because when HDMI CEC is on, the receiver should automatically switch to the HDMI input that powers on and take audio from there instead of statically preconfigured one.
If your device does not support HDMI CEC (most cable boxes don't support that), you should not use HDMI control in the first place.
As suggested, a universal control like Harmony is a better and reliable solution. I always turn HDMI control off.