I also found the message alerts etc don't come through the car speakers when using bluetooth.
Calls definitely come through and that works well but alerts don't come back until it is unpaired with the hands free.
Even my very old Nokia puts all alerts through the speakers in the same car with the same hands free system (built in) so what is wrong with the IPhone
On another thread someone said you should not read texts anyway in a car but you don't, Siri can read them to you but you need to know you have them before you can ask Siri to do that.
You also need to know you have received a message if you want to pull over and stop to read it.
As a business user who spends a lot of time on the road this is frankly an appalling situation and tempts me to go back to the old Nokia.
All the Siri technology etc and they stuff up something as basic as this then I find out it has been discussed on this forum for a couple of years.
All it takes is a software settings on/off option "rout alerts through blue tooth when connected" yes/no