There's a lot to be said about that SYNC unit itself.
Here's a little more to help you decipher what I said above.
If you remember the older Console Component stereos of older days, the SYNC unit takes a few pages from them to enhance your ability to be entertained while driving.
The older days of component systems used a "tape monitor" loop - it allows you to insert specific components to help process the analog audio thru the Stereos Line-input and output level - so the system can add flexibility to let you hear not only the PHONO or AM/FM Stereo - it allows you to add in Reel-to-reel onto Cassette and 8-track or even use imaging processing like dBX and Omnisonic audio processors to make the music and the musician vocals appear live and in-position as if you were sitting right in their studio with them.
SYNC and the console Audio Head unit, are set up so they communicate to each other and the SYNC unit also processes other inputs into a means that the Head unit - the one behind the Dash, can play the music, MP3 or as you hope to have, Bluetooth so you can connect devices to further enhance and exploit the devices' you connect to make your trips easier.
Well, "Easier" is subjective, but the idea to handle phone calls and listen to a better selection of music you want to hear - is part of the reasons for SYNC was designed to take over that "Tape monitor" loop the Sony (or any Ford Head unit) so it can equalize, the BT-connect, MP3 player decoding and selection and Satellite along with the many AM/FM channels present in the airwaves.
So when you lose BT - the BT unit is on the SYNC head - not at the SONY side - but to "Grey out" or no longer offer BT Mode On - then that tells me the SYNC unit is not working - as in - no power to it.
Where SONY head unit comes in - is you don't have any Player - volume On/Off or and channel selection - when you don't have that - then the Head unit is showing it's age.
There are several ways to try and reset-or-restart the SYNC unit. Mostly from the main Console menu where you Select Factory Default or you can even try thru PHONE menu a reset all or Erase All - to hopefully let the unit re-pair to the phones and devices you have - but that is under PHONE menu on the Console.
But if you can't even get to the PHONE menu - then remember too, that the 3-G / 2-G support that 911 once had that SYNC was set up for - is no longer available and is considered a security risk for some unscrupulous dealer or even a driver alongside your car, can transfer data to and from the SYNC unit using BT to attempt to pair as a similar device thru image copying the ID they can read by looking at your BT profile of either your phone or SYNC itself by looking for devices that are currently visible - like you can too, so you can select the device for paring - so remember to stop the scan your phone makes to look for other devices once you've been paired with your sync..
It's kind of like "Phone slamming" where all the operator of the older days had to do, is call your number - and if you had an Answering machine - if it'd pick up - that's all they needed to convert your phone to another plan - the pickup of the answering machine was the trigger to tell the system connection established - and so the call was completed and you didn't even know it happened until you look at your new bill or lose service because of the biller didn't receive payment because of the switching to another provider - you now have got no phone service and a big bill - possibly from two plans.
The FTC took care of that, but in the BT realm, this can happen more often than you think - even FitBit units are having their data stolen so the "steps an calorie" counts data and profiles can be sold for profit. Walgreens, CVS and others use your steps data to help you keep well and offer wellness plans - but they don't necessarily look at whose data it's comes from to get those free points and stuff - so these dealers with their apps "steal" from the fitness trackers as they can to sell this info for this purpose.
So in short, the SYNC unit may be the whole mess - and this forces you to take a step back and reassess your needs for; is this type of operation necessary or would it be easier to use a more committed system and keep the Phone as a Phone and just use a simpler AM/FM car radio in AUX input and just leave the phone alone. The problems with SYNC units are their need to be in the system to handle upgrades that no longer are available so the Head unit and SYNC itself don't or can't be upgraded by chipset or by software - that support is now gone. And with VIN requirements like Ford GM and pretty much any other car maker now - all require the notion and enforcement of;
If it's not on the VIN (your Seller sticker on the car)
it's not on the Car - at least thru us.