It would be completely different story having the lithium battery pack connected as the main and ONLY battery and cranking with it four times a day.
A typical car battery can begin at 55 Ah 12 V, this is for a European downsized petrol engine (larger engine requires more, diesel requires more, electric accessories require more) from before the advent of start-stop systems (start-stop requires better batteries or even two of them).
That is 660 Wh.
I guess the weight and volume of this in li-ion would be similar to or slightly better than lead-acid (imagine roughly 10x laptop battery or 75x 18650 cell), but the cost would be a multiple even at manufacturing prices.
Next, the car battery can sit in freezing cold for several days to weeks at -20 °C and still crank or it can be frying itself all day under direct sunlight. This is deadly for li-ion. Electric cars have heating and cooling systems for the battery packs even when the car sits still in a parking lot.
Also, while driving, the battery sits next to hot engine. You can move the battery elsewhere (driver seat or trunk, like the second battery for start-stop systems), but you need high-energy leads and the engine heat is actually a benefit for recharging the lead-acid in winter.
Compare all these complications to price, simplicity and reliability of nicely recyclable lead-acid box...