This company builds energy storage systems that use hoists and heavy bricks.
We’ll stop you if you get too technical for us. 
Though seriously, it is interesting that many of the solutions for one of the major infrastructure challenges of our times are actually really simple to explain.
Which means that you don’t even need to store the energy.
And the broader the range of comfortable temperatures you’ll accept from your house climate control system, the more efficiently you can make use of intermittent power sources.
I have spent my entire life living on the coast where it is cool, few people even use a/c. But the past 5 summers I have spent in the desert. ( fairly dry heat) I find I seldom use the a/c. We are pretty adaptable if we allow ourselves to be.
My understanding is those ice storage systems are really meant for a place like a data center where cooling is always needed. If the overnight electrical rate is lower, the ice can be formed then and melted during the day to cool the computers.
Most of the docks in the UK once had hydraulic power systems, which used water pressure to power capstans and other machinery. These hydraulic systems were powered by large hydraulic accumulators, basically heavy pistons housed inside large buildings, which were raised by steam power or some other form of energy, and thus retained a large amount of potential energy for ready use.
Similar systems could be used as gravity batteries in the current era.
having investigated a similiar (but way more basic) system as the ice example to cool my house in summer…
the main worry is: condensation of warm and humid air on a cold structure (think cold beer-glass in the summer) … with potential for mold/dripping/structurally compromizing concrete/rust…
so there need’s to go a lot of thought into that - and quite possibly de-humidification
I live in a similar climate (warm/hot in the day, pretty nippy at night)
your ticket for good quality interior climate is good enough insulation (sip-panels in the roof) - and solar mass (bricks/concrete walls).
Warming the brick structure during the day, to radiate warmth in those cool nights - and starting the day out already cool to keep the interior also cool. So, basically just inverting the ambient cycle.
those daily cool/hot cycles are actually pretty easy to work with and to use them to your advantage.
Gravity “batteries” may not be the optimal long term energy storage.
There’s a new operation near Pahrump NV that stores energy by hauling heavily loaded railcars up steep ramps. Ten of these ramps, I believe. It’s built in an unused part of a quarry, so it’s not chewing up pristine nature. You’d think it’d be able to hold lots of energy, but its capacity is only 12.5 MWh. It’s a 50MW system, so at max power, it’ll discharge in only 15 minutes.
For long term energy, such as storing energy in summer for use in winter, we need many gigawatt-hours of energy. Gravity batteries like this one just aren’t energy dense enough. I suspect that applies to most, if not all, gravity battery operations.
There are many different timescales that are relevant for power demand smoothing, with different technologies suited for them. It might be that the 15 minutes you get from a plant like that is enough time to start up some system that is slower, but with higher capacity.
Ideally if you lifted something 5,000ft you would be bringing something up with the load that was needed up there.
That’s not an energy storage scheme, though, because if the load was needed up there, then you wouldn’t be able to lower it to reclaim the energy. If your goal is energy storage, then you’re probably just loading up the train cars with something cheap, like rocks (especially since the project is sited in a quarry, anyway).
Though if you have some industry that does get some use from lifting up loads (and leaving them there), you might be able to set it up such that you’re only lifting when energy is cheap.
speaking of which:
another thing that is done here locally IIRC (northern Chile, mining-country - where all the mines are pretty much up there at 4,000+m above sea level and the ports are pretty much at sea level).
Braking of the full with ore (=heavily loaded) rail cars - with a 4,000m “head”, creates the energy needed to bring the (light = empty) rail cars up from port to the 4,000+ meters of altitude “for free”.
that makes a LOT of sense if you think it through