I actually spent a few days last week trying to solve the same issue. I'm a delivery driver, so I'm always using my phone in the car. I was very disappointed with the few car mode apps that there are, so I made something myself, and it's been working flawlessly
After trying a few things, I figured that short of building an app, lightning launcher was my best option. In LLX, desktops don't mean like scrollable pages in most launchers, they're separate entities like multiple desktops on OSX or GNOME. So I created a wonderful looking portrait desktop for normal use with LLX and Zooper, and another landscape one for car mode that matched my theme, and would become my default desktop when my car is connected. I have phone, mms, voice search, Waze, and a big clock on the main view, and I can scroll left to view a full screen Zooper widget with media controls and album art, or I can scroll right and view direct dial shortcuts for work.
So when I start the car, BT connects and tasker automatically switches LLX to car desktop, disables display timeout, turns media volume up, and brightness up. As soon as I shut the car off, BT disconnects and it switches back to the portrait desktop and old settings are restored. I don't have to press anything, and I don't have to run an extra app and worry about car mode. I'm not experienced with any of those apps and it looks and works perfectly. Took a few hours of playing around to get it just how I wanted it, but I couldn't be happier. If you have any questions or want details, just ask!
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