The problem with all of this is that you would need to swap SIM cards on the cellular-wifi router almost every time you change locations -- and that's if you can even find a data vendor willing to sell you a prepaid data plan without a credit card on file.

Practically, this solution you've come up with is too fragile to be relied upon. It's only secure if you can maintain this level of care with every operation you perform on the phone. This is why the NSA's data collection is so insidious -- it only takes one slip-up to connect everything you ever did anonymously to your "profile".

With the type of data collection that is done today, it's nearly impossible to avoid unless you use the Internet in a very different way than the average person. Avoiding data collection through technical means is a futile exercise at this point -- if you object to the data collection, public policy is the best avenue to prevent it at this point.