Since they don’t come with music Bluetooth compatibility only for phone calls

For context: I modified a car stereo that didn't have an aux port to have one, it worked and everything was copacetic. One day while driving the volume on my phone started going up and down randomly and the songs started skipping forward and backwards in the list randomly. I narrowed it down to a faulty pin inside the 3.5mm jack in my phone that, with next to no probing, proceeded to just fall out (its an old phone that I've plugged/ unplugged headphones very many times so this kind of thing was bound to happen sooner or later). Luckily the audio playback still works but now I can't make phone calls hands free either in the car or with headphones while I'm out of the car. I tried to find a solution and I've bought a few different products from the source/best buy and only one sort of worked except even on full volume on both phone and stereo, the sound was super low and barely audible, and normally just through aux I can play music off my phone at a very loud level at just under a third of the way on the stereo with the volume maxed on my phone. I've come to a point where I'm not sure if I'm looking for the right thing or If I'm going about it incorrectly. I know for sure I don't want an FM transmitter (a lot of the people at the stores kept trying to sell them to me even after I told them repeatedly that its not what I'm looking for, ending up being reassured that it is, buying it and walking to my car, trying it then walking back and then having to return it because it didn't work. I think what I'm looking for is a blue tooth receiver but I'm not sure as the ones I've tried haven't worked.

What I'm looking for: I'm looking for something that I can connect to with my phone via blue tooth that outputs an audio signal via a 3.5mm jack, with a built in microphone so I can make calls hands free again/ when I'm not in the car be able to use headphones while on a call.

Added a janky MS paint picture with some text to hopefully help with what I'm looking for

Also, generally if my car is moving then beyond hitting next song in my phones music app or swiping to answer a call I don't like doing anything else like texting or making calls/ going through apps/menus unless I come to a stop that's not in traffic, I try to avoid making calls unless its an absolute emergency, but having to switch to speaker phone while answering calls if I'm moving and not being able to hear the other end/ them not being able to hear me as well as not being able to make a call while having my hands free while out of the car is/will be fairly tedious until I can get some money saved up for a better solution like an aftermarket stereo or new phone/fixed phone.