Let me share how the Logitech Revue used that HDMI input

It basically would switch to that input at times. Tuning was done at the cable decoder so it was of little value.

In the current setup it was even dumber to have the HDMI input on the Revue because we are using the coax connection on the TV.

So to watch TV we use the TV remote to select which source and channel.
To use the Revue (or this Sony) we would use the TV remote to select the HDMI source and the Logitech or Sony remote to work that box.

You don't want to convert your cable to HDMI. Dead end, full of sorrow and woe.