I think your three objectives here, 15-20 km days, not to book ahead and not to have to walk on another 5 or so km, are almost mutually exclusive. I walked the CP from Tomar in May 2022, and used a combination of booking ahead a day or two, ensuring I arrived sufficiently early where I didn't, and choosing places where there were several albergues and other accommodation closer together.
Here is what I see as the rub. If you plot out out 15-20 km stages (I used ~20km stages) you find yourself stopping more often in places with only one albergue or hostel. This increases the prospect that if you haven't booked and it is full, having to walk on a little longer. How often is difficult to say. I only observed it once, but I wasn't sitting watching the arrivals, and where I did see it I just happened to be in the reception area at the time.
Where I arrived early afternoon at places that I hadn't booked ahead, I never had a problem getting a bed. But at the same time, when I passed albergues in some of the more popular towns, there was already a big crowd waiting by midday or shortly after. Perhaps not enough to fill these places, but certainly enough to make me concerned.
Your question hints at using taxis should you find yourself in this situation, so clearly you understand what is at stake here. That is not something that I had to do. Certainly there were taxi services all along the legs you are proposing, but I have no idea how much they would cost, nor how responsive they would be.
On your last point, I found Gronze.com to be most useful in finding accommodation. I did have the Wise Pilgrim app, but didn't use it regularly to navigate, and didn't explore its potential as an accommodation search tool. Looking at it now, it seems to be a pretty good complement to Gronze or as a standalone tool.