Lead has a low solubility ...

Lead has a low solubility but that is often sufficient to exceed the acceptable standard. This is made worse by low pH and low alkalinity water so that many groundwaters have a low tendency to dissolve lead but surface waters may be much worse. Also the time in which the water is in contact with the lead  pipe is a factor. This is why there is often advice to flush systems that have been standing say overnight before use.

One widely practiced technique is to dose phosphate to the water which forms a protective coating on the pipe surface.

One problem is that a lot of testing is done with coupons in tanks of water that does not reflect the pipe situation. That is why a lead pipe test rig is a valuable tool to check the risk of particular water works..