The "need to know your position" is a consequence of how they measure signal strength to allow handoff between cells, so you get overlapping coverage and each tower has a signal measurement for the same device.
A post I saw here a couple of weeks ago suggested a simpler method of reducing location tracking than ones I had been thinking of:
You would add extra RF spectrum overhead and a little bit of latency for making and receiving calls.
In general the basic thing that should happen to recover location privacy in mobile services is completely decoupling payment, device identifiers, and routing. Right now they are all linked or trivially linkable (your carrier sees you register with your IMEI and IMSI, and it assigns you a phone number -- rather than registering with a random per-session IMEI, paying for the service with a blinded digital payment token, and getting a phone number, if you want one, from a third party VoIP provider).
Great. Here's your next phone. It puts out 2,000 watts and comes with a free tinfoil helmet. Battery life isn't great, but at least you're not being tracked by local cell sites.
edit: though I guess that fact that you are connected to a specific tower leaks location information