I've often wondered if leads ability to stand a fairly high c rate for short durations is not simply the mass that has to heat before the cell gets all hot and bothered. The engine starts before it can heat all that lead much. It does of course, also depend on plate thickness. Starting batteries can punch it out because of the thin plates.
30 pounds of lead, vs 10 pounds of lithium foil and carbon sandwich, the lithium could overheat and get damaged a lot quicker I'd think.
But for continuous discharge, like EV's do, lead does not shine. It really has to stay well below 1c in my opinion, or it only lasts 6 months. That's part of why those lead powered stand up scooters tend to be less than 500w.