If possible, make the device connected to HDMI turn on the TV via CEC, that should switch it to the correct input automatically.

That's strange. I have an LG (a couple of years old I think?) and it defaults to whichever input it was on previously.

So if I turn it off while on HDMI 3, it turns on to HDMI 3. If I turn off to antenna, it turns on to antenna.

Only irritation I really have with it is that it's run into the odd error and will reboot suddenly. :|

I dealt with getting blasted with the "No signal" thing on my LG webOS TV for several months before finally figuring it out. I wish I could remember how and tell you, but I can't recall. But don't give up! It is possible.

EDIT: Actually, I'm not sure if I got it to default to HDMI but I did definitely get it to show something other than no signal. Try using the picture frame feature (where it shows art in a picture frame). After so doing, I think it at least defaults to the picture frame. I also have a Roku connected that seems to be capable of telling the TV "hey switch to me", so sometimes I get the Roku HDMI and sometimes I get picture frame, but at least I never get No Signal anymore.

It should default to showing whatever input is currently marked as "TV/Cable". Out of the box, that is the OTA tuner. If you want it to default to an HDMI port, you will need to use the connection manager to tell the TV that there is a cable box connected to an HDMI port and to use that as the TV input.