Thank you for your reply Eric,
1, I really would not upgrade IOS anymore as it slows down my device. I have had this issue since maybe from IOS 14.0 or even 13.5. I cannot really remember the exact version because in the past I have always set the Auto-Lock to "Never", therefore I could not notice the issue until I set it to "30 Seconds" later on at some point. However, once I found this issue, and if there is a new version of IOS, I upgraded till now to 14.5.
2, About "resetting the IOS", I have done it a very long time ago and many times, and I am not sure if I tried "Erase All Content and Settings", but I clearly remembered I definitely tried "Reset All Settings" on my phone.
3, I get in the car and paired the stereo via Bluetooth so I can play music right? When I was done and get out of the car it disconnected from the car's Bluetooth right? And then occurs the issue. As for "What happens when you press the Lock button on the side of your iPhone while paired with your car stereo", it just locks the phone, nothing weird happens.
4, I have not another car to test, but it does not happen with other Bluetooth, like headphones whatnot. My wife uses iPhone 11, and she does not have a problem with paring my car's Bluetooth.
I know what I need to do is probably "Erase All Content and Settings" to set it as a new device and upgrade it to the latest version of IOS, but it just too much for me right now. My content is huge, very time-consuming, and my device is too old to upgrade.
I guess what I want is to report this problem to Apple, giving them a heads-up. If it is valid, maybe there is something that needs to take a look at.
I definitely will "Erase All Content and Settings" to test it out, but it will be the time when I am getting a new iPhone.