I spend a lot of time in libraries on public computers or on public Wi-Fi. Their policies often state up front that there is no expectation of privacy, that staff can check up on your activities if they have reason to do so. I have not paid much attention to policies at, say, Starbucks, but I wouldn't be surprised if they have similar policies. Furthermore, my understanding is that Wi-Fi has pretty big security holes compared to a landline. Plus, if you are a regular, people will recognize you.
I gave up driving years ago and I walk everywhere. This is bizarre and noteworthy behavior in the U.S. People stop me and talk to me and say "I see you walking All The Time..." The degree to which people in cars notice me, recognize me and feel not only free but compelled to speak to me is downright creepy.
So I will suggest that if you spend time very regularly in cafe's using their Wi-Fi, etc. people will not only recognize you, they will feel friendly and curious and like they have some goddamn right to grill you about your life and why you are there all the time and so on and so forth.
I also agree with AnimalMuppet that the lengths to which you are willing to go in order to be "completely anonymous" raise enough red flags that someone, at some point, will take an interest in tracking your ass down and that "someone" may very well be a government agency. So while I get annoyed at how humans are wired and how they conclude they have some goddamn right to grill me merely because they fucking recognize me when I have no clue whatsoever who they are, beyond being annoyed as hell at the whole thing, I don't really need to worry too much because walking everywhere isn't actually a crime, no matter how bizarre and eyebrow-raising it is. But I cannot imagine any reason to go to the lengths you want to go that don't involve serious crimes and most other people will be far more critical of your motives than I am. I am pretty live-and-let-live. On average, other people are much more judgey, butt-in-sky and controlling than I am. So you can bet dollars to donuts that most people will assume you are up to something incredibly evil and that suspicion will fuel their interest in grilling you, tracking you down, etc.