Stereo stops working

You might want to check the repair Manual or just find the under dash fuse block for the location of the radio fuse. A youtuber with a 2003 had the same issue, but had a hard time finding the blown radio fuse.

Or you can read this 3 year old thread for all the Dealer horror stories about the same issue.
Audio Not Working Mystery - 2010 Prius | PriusChat

Good Luck and I hope it's only the fuse for you

 

I'm not as familiar with Gen3 as I am with Gen2, but is it possible the volume control buttons on your steering wheel are malfunctioning?

 

Intermittent on / off failures are one of the hardest issue to troubleshoot. It may take a while to find the problem and I'd keep a diary of when and how it started and what you've done to try an fix it. ie: does it happen every time you use the audio system, when it started happening, etc...

Trying the easy stuff first is usually how a DIYer will tackle issues like yours.
I'm just googling 2012 Prius audio issues If you want to look at what I'm going by.

Next easiest thing to try, since I really have no clue what is causing the issue, would be disconnect your negative 12 volt battery cable. Be aware when disconnecting the negative cable you will loose some of your personal settings.

You could also try disconnect the negative cable, pump the brakes and / or let the car sit a half hour disconnected. Than see if the issue continues after reattaching the negative cable.

 

Is there any damage to any volume buttons or the knob? If I were in your situation I'd focus on those. Loosening them up/exercising them... There's also a function in some stereos that mutes when there is an incoming phone call or when the settings are changed...

Another option to consider is hang out in a car stereo store parking lot near where the installers work and when one of them has a spare minute ask them if they or their co-workers have ever had this problem and how they fixed it.

And if all else fails, it's a good excuse to upgrade your sound system...

 

I imagine the reason the dealers say a new unit is necessary in a lot of cases like yours (as seen in the priuschat thread I linked above) might be, in a lot of repair manuals, many times it's recommended that the technician replace the failing unit with a unit that is known to be a working unit (aka: a new one).

Sometimes, in the case/s of loose or bad connections in the wiring, simply removing the unit, unplugging and plugging each connection again can fix an issue like yours and many others besides.

I know it is a fairly large undertaking to tare apart the dash and remove the unit with all it's connections, but you'd most likely have to be willing to learn how to do that yourself, because most professionals would need to charge a hefty price to perform that kind of service.