I know you probably won't see this, but I made an account just to thank you for this!! I had the exact same problem, my remote would lock/unlock doors with the buttons but the keyless entry and starting wasn't working. I scoured Google and most places said the remote had to be reprogrammed. Finding your comment was a needle in a haystack! I did exactly what you did, took out the panel with the HDMI and lighter ports, and there was one cable disconnected! Plugged it back in and like magic, everything was working again.

I just had my car worked on (mold and water damage to the floors/carpet, blegh) and a fight between my insurance and the shop meant 2+ weeks of car stress. When I finally got it back and the remote wasn't working, I really didn't want to have to take it back in. Your comment saved me! The shop must have overlooked that one cable when they were cleaning or whatever.

 

While I'm familiar with those cables from my time adding an amp and upgrading my speakers on my Fit, I find it really odd that those cables being unplugged would cause issues with starting.

 

There's one specific one that is on top of the HDMI and cigarette lighter plugs, it's weirdly horizontal where the others are obviously coming off of where the plugs are. I almost couldn't find where to plug it into. My guess is that it powers the receiver in the car that listens for the signal from the remote?