"Dumb phones" may very well have the same software hard-coded without you knowing. It's much harder to tell than with the smartphones...
But in a way it does not matter if your hobby phone is smart or dumb and is recording this data if it has no link back to you and is not turned on at home much (since if it records rough location it could determine your home was where it sat still the most time).
I have an "Idiot Savant" hobby phone, that is not a modern smartphone but does have a keyboard to make typing texts easier. I assume every one of those texts (incoming and outgoing) is recorded by the phone company. But since I have the phone on only rarely and it's an all cash phone, it doesn't matter.
For the really paranoid, did you know the cell phone company can remotely turn on a phone anytime they like and then track down its location? It's almost never done but in theory with the right warrant LE could ask for that. So you may even want to consider a hobby phone with a removable battery... again only if you are REALLY paranoid.
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