It depends on what kind of privacy you are talking about.
If it's about the carrier/government knowing all places you go, when you are moving, when you are not, where you live, whom you call, whom you text, who calls you, it does not change much: triangulation will show your coarse position on a modern smartphone and a dumb cellphone the same way, and the call/text metadata is still available for the carrier and government, and this data does not depend on the phone.
It's almost like running MS-DOS today: you cannot get a tracker written on .NET to run. You cannot exploit a drive-by download vulnerability on it. You cannot exploit SAMBA on it. Those vectors won't work because the attack surface lacks a lot of components.
It's better? Sure!
It's perfect? No. You can still be tracked, in a more limited fashion, but can be tracked.