My car runs off of a 3.3lb LiFePO4 battery. It has very little capacity compared to the 50lb lead-acid battery it replaced, but has no trouble sourcing enough current to start the car reliably. Lithium batteries have the ability to source tremendous amounts of current, which is why they're so popular in RC models. The best RC lipo batteries can source 65 times capacity continuously, and are often limited by the cell tabs, wires, and quality of solder rather than ability of the cells. LiFePO4 doesn't have quite the same power, but it's still more than enough to start a small car engine.

I think the reason why lead-acid is still common are:

1) WAY cheaper per capacity, by a TREMENDOUS margin (10x+).

2) Better low-temperature performance.

3) Better tolerance to being abused (overvolted, run dead).