My Left Stick Click Has Stopped Working, L3 Does Nothing At All

You have diagnosed it correctly and I can save you the disappointment of opening it up hoping for a separate switch.

On a DualSense, the stick click is a small tactile switch built into the base of the stick module itself. It is not a discrete part sitting on the board that you can swap on its own, so a dead L3 means a module replacement. The upside is that this is exactly the same repair as a drift fix: same module, same part number, same price, same procedure. If you have ever read a stick drift repair guide for this controller, you already know the job.

It is worth understanding why this fails independently of the stick's analog function, because it reassures you that the rest of the module is irrelevant to the diagnosis. The click switch and the two potentiometers are separate mechanisms inside one assembly. Your potentiometers are clearly healthy, since your drift and deadzone readings are good. The click switch is a dome contact that takes the full force of your thumb pushing down, and after two years of sprinting it has either worn through or its dome has collapsed. Nothing you do to the analog side affects it.

One thing worth doing first, because it is free and occasionally works: the switch can be held open by debris or by dried drink that has run down the stick shaft. A few drops of isopropyl alcohol into the stick base, then click it forty or fifty times to work it through, sometimes revives a switch that is contaminated rather than worn. It is a long shot at two years of use, but it costs ten minutes and the module will still be there afterwards.

And a practical note while you wait for the part: remapping sprint to a rear paddle or to Circle is a perfectly reasonable interim fix and most games allow it.