But fossil fuels do also solve the problem of storage due to their energy density. One reason you put gas into a gas tank is not only is that a source of energy, but it stores a lot of it in a small space. The tank is also just a cheap steel container that whose capacity doesn't diminish with number of refills.

If you can replace that with a battery with all the same properties: cost, space, weight, durability and so on, that does have an impact.

The energy for filling that battery can come from fossil fuels, but doesn't have to. We have hydro, wind and solar.