I got a new bike about a year ago and have had no back pain since (until randomly yesterday, but I'm 41, I guess it's expected that it'll happen occasionally).
Biking doesn't seem like a back exercise, but you are using your core and flexing it a lot, and for longer than you would on any standard core exercise.
Building exercise into your day to day life is incredible valuable. If you're lucky enough to be in a place where biking is viable, I highly recommend giving it a try. It's not for everyone, but given the right situation, I suspect the majority of people would find it a positive change to start biking more. (I suspect this is true for a lot of exercise, but biking has the benefit of being a productive workout—similar to chopping wood vs. lifting weights, except few of us need much wood chopped.)
Unless you have one of those lounger bikes, you're getting an exercise that hits almost everything below your ribs.
https://www.huffingtonpost.co.uk/2015/10/12/cycling-add-year...
1) https://www.vox.com/2015/3/19/8253035/roads-cyclists-cars-hi...
I really hope US cities head in this direction. It is a no brainer for cities: 1. higher density -> lower taxes per person, higher tax rev per sqm 2. lower cost of living -> lower inflation 3. quieter; cars on streets are noisy 4. people see each other more; better social cohesion 5. kids are free from the house to do things