I think the challenge with your plan is it requires potentially a lot of walking. If you walk to one place and it’s not taking walk ups or the line is very long, you may have a long walk somewhere else. As well, rides are often down these days, sometimes for hours at a time. Without a phone, the only way to know that is to walk to the actual ride. Again, a lot of possibly “wasted” walking. Now if your attitude is that if we walk there and the ride is down/wait is too long/no walk up ordering and you are ok with just the enjoyment of walking there, then you will be fine. It would not be ok with how my family does DLR, but it might very well be ok with yours. For what a day ticket costs, I would not be ok with that plan, but again, you might be.
Even if your phone doesn’t support the latest version of the app, can you see wait times and do a mobile order from the version that your phone does support?