Nice to know. I retrofitted SiriusXM in my TMY, so I have the satellite-based app (I have 2 lifetime accounts), and pay for streaming audio with LiveOne, so I haven't needed to try streaming via the web browser. Thanks for the update.
As for Audible, I'm not sure, short of using the bowser instead of the app. One thing to note, there is a known issue with connectivity when first starting your Tesla. If you have wifi where you park, it will join, or if you tether it will join, but it will disconnect when you put it into drive. One of the later OS updates allows you to set your connection to remain with Drive. If not it will disconnect. This is an issue with people who use MyQ for their garage doors and have premium connectivity. There ends up being a time where the vehicle loses connections and can't control things.
One thing may be to switch to a different audio source before you park, so that when you first get in the vehicle it doesn't start in Audible. Once you get driving a bit, switch to Audible and see if it syncs properly. Not really a solution, but something to test to see if maybe it's starting to play before the app even tries to sync.
This is where the argument of having CarPlay/Android Auto has merit.
The last failsafe is to simply use BT and stream from your phone. Since you are listening to an audiobook, it shouldn't require interaction with the phone for most, if not all of your drive. So, once you start it you are good to go, you can pause and play with the Tesla UI. For me, I always have Waze running on my phone (with the setting to send audio through phone BT instead of Audio BT), so I get audio notifications like speed/redlight cameras, police, traffic,...