But you haven't bought one yet?
It did cross my mind too but I don't really do anything where I would need one.... yet
Have you looked into one of those phone companies that is supposed to have software that no one could track? Never sure whether I actually trusted that they were secure and not just a trap.
My privacy would only be critical if I were trying to fly under the radar. Until then, they can waste all the time they want gathering digital metadata on me. I lead a very boring life -- one reason I never had a problem getting my TS/SCI clearance.
The only part of my life that might raise suspicions is if an investigator can't believe anyone can be THIS squeaky clean!
Not sure which software you refer to. Every product has its strengths and weaknesses. Nothing beats a point-to-point fully encrypted path using a dedicated satellite channel.
I looked into the Walmart phones years ago out of curiosity, and the cost wasn't bad. I have AT&T, also, and looked into the pay-as-you-go plans for additional phones for visiting relatives or as an emergency spare. Now, I just swap SIM cards to an old phone if I have to have mine repaired.
Overseas, you can buy SIM cards with prepaid minutes -- no identification needed. Unfortunately, the phones available here don't seem to be as simple to find and buy. I suspect the gov't is trying to keep untraceable phones out of criminals' hands, like drug dealers and so forth. It's really difficult to find a SIM kit or disposable phone that you can activate without giving your real identity.
I know there used to be big (illegal) money in stealing cell numbers and ID codes from the air and reselling them. The info allowed cloning legit, active cell phones so you could use their service for free and have no record coming back to you. Haven't seen much of that in the news for years, so the cellular providers must have ways to prevent it now I'm speculating.
If you watch movies like the Bourne collection or the series 24, you'll see them destroying phones, SIM cards and removing batteries all the time once they've used a phone. Jumping from one phone to another is about the only way to never have a phone, even a burner, traced to you. I remember a Law & Order episode where they found a burner phone in a guy's apartment that was used to call another suspect. The detective said, "They call these burner phones for a reason. You're supposed to get rid of it after you use it, Genius!"
So, having a burner as needed makes sense. Use it and lose it. Then get another. It's the only real way to be safe. Kind of like not hanging onto a murder weapon. If it's ever found on you, it can connect you to the crime.