These measures in the U.K. were put in place to shift demand due to constraints in total system capacity - total generation, total transmission, total distribution capacity. Eg a substation transformer couldn’t handle normal daily loads plus heating, so heating load is shifted to night.
What happens when power at night becomes more expensive due to increases demand such as car charging or low supply such as high renewable penetration and no wind?
It’s all about having the firm capacity to deliver power for hours on end. So the $/mwh of wind or solar power has to go up because grids are going to say you can only have more if you can supply firm capacity. This is already happening in Hawaii.
I implemented hot water tank load switching through smart meters on at a town with its own power plant of only 850 kw so they wouldn’t run out of power at dinner time when the river was low (and we didn’t have to fire up the diesel either)
One way this could work is that you have eg a smart heating thermostat which plans which hours of the day/night to turn heating on and then buys power to cover that in a day-ahead market, with the aim of minimising cost on that basis.
Wind and solar power are relatively preditable a day out so you wouldn't be paying too much to power traders for providing day-ahead liquidity vs on-demand pricing.
Likewise, EVs all end up parked somewhere during the day - the challenge is ensuring that as many of those places as possible (car parks, workplaces, etc) have charging points too; you could do something very similar there with an algorithm deciding whether to charge based on a spot price and how much is in the battery (say, owner programs car to top up to 65% and then only continue charging if power is cheap).
The beauty of this type of approach is that it can be done by incrementally building on things which already exist.
There's already an electricity market, including day-ahead pricing in most pools. Professional power traders already trade these markets. Smart meters already exist and are capable of dealing with pool prices not just flat rates. Smart thermostats already exist. EVs already have the capability to stop/start charging based on more than just whether they're plugged in or not. Even better, most of these already have standardised APIs.
What's missing is the software stack which ties all of these together - and that can be done incrementally, device by device (on the load side).
People who have those tariffs do tend to alter behaviour to suit them - it mainly makes sense for homes with electric space and water heating, where it's more "set and forget" (and people also do change behaviour by eg putting washing machines or tumble dryers on late at night when the cheaper tariffs apply too).
Over time I imagine we might see more dynamic algorithms in use to pick and choose times when electricity is used together with more dynamic per-unit pricing.