The cost difference depends on number of cycles. If you fill and empty your storage daily, batteries are way cheaper. If you fill and empty annually, pumped storage is way cheaper.
Hydrogen is a very poor long term storage. It either uses very expensive tanks or it leaks away. It's also a couple of orders of magnitude more expensive.
No, hydrogen is excellent for long term storage. It is stored underground.
"Couple of orders of magnitude more expensive" is not correct. For long term storage, the capex per energy storage capacity dominates, and the cost of underground storage caverns (especially solution mined in salt) is very cheap, an order of magnitude cheaper per unit of stored energy than reservoirs for water.
It absolutely does matter. If one is cycling the storage system annually, $3/kg for hydrogen becomes a minor part of the cost. The cost of this input is proportional to the number of charge/discharge cycles over the timescale indicated by the discount rate, and for annual storage that's not very many.
It's a common mistake to think that the importance of round trip efficiency for diurnal storage carries over to annual storage.
