Recharge rate is not an issue where there is conduction through rock, from a higher temperature energy source. Rock is a good conductor of heat. Any energy you take out of heated rock will immediately be replaced from the higher temperature region adjacent. It's the second law of thermodynamics. Heat always moves from hotter to cooler regions, and passes easily through stone. — counterpunch
This is simply not true. On the scale of literal insulation being a good insulator and a heat pipe being a good heat conductor, rock is closer the insulation side. Energy flows from hot to cold, but it takes time.
Indeed, the reason you need to drill to the depths you're talking about it because the rock is insulating pretty well. If you drill to those depths and extract heat from a small volume, it will indeed recharge pretty fast, but the larger the volume the slower it will recharge.
The volume of rock you need to power something significant (like a continent) is completely massive. To justify the capital equipment of the power station, not to mention the hydrogen production and storage and transport terminals you've been talking about, the amount of power needs to be "worth it".
Which is why, as
has posted, many wells fail even in super sweet spot regions such as Geysers that is "the most developed", 15 wells produce 725 MW. An average of 49 MW per well.
That's just not a lot of energy. If heat just "instantly" recharged the rock at 700 C, you wouldn't limit your generation capacity to 49MW per well, you'd just "let her rip" and have nuclear gigawatt power station sized pipes. The reason, once you have a well, you can't just circulate as much water as you want to get as much energy as you want with the rock heat recovering "instantly", is because it's not instant. You're energy extraction must be equal to the recharge rate of the rock volume you're extracting energy from, otherwise the heat source cools to a point you can no longer generate power. Indeed, many geothermal stations end up running at a fraction of their original design capacity, because the recharge calculations were wrong.
The volume of rock will cool as energy is extracted from it, but will only heat up proportional to the surface area below (and to the sides somewhat, but doesn't change the proportionality here).
When you have these sorts of volume to surface area proportionality constraints, the solution is to have a small volume to keep that proportion low; hence stations are in the single or double digit MW range and not the GW range.
Another way to look at it is the heat gradient from the core to the surface. The earth is efficient at trapping all that heat: i.e. the opposite of being efficient at bringing it to the surface; we can make it more efficient by drilling down there and circulating water, but once we do that for a large volume we are constrained again by general heat gradient context of the surrounding rock.
Solar energy does not have this problem, but the energy extraction is a surface area (PV or hot water panels, mirror for solar concentrates, or just windows to heat buildings) and the "recharge" rate is proportional to the surface.
That the energy is spread out over the globe just means it's its own distribution network and we don't need that capital cost.
However, I am not arguing that solar energy will prevent the the climate change catastrophe. I am arguing that it could have, if fossil costs were internalized in the 70s - 80s - 90s, but now it is too late to avoid major tipping points.
The Amazon being a carbon source now, instead of a carbon sink, is a major such tipping points (I remember being discussed literally decades ago as a "oh shit moment" we should try to avoid). Likewise, that temperature records have been recently broken by several degrees, is also evidence of the climate breaking out of the meta-stable Holocene epoch, but the entire Quaternary geologic period, and is currently in an unstable region that will move rapidly towards a new metastable point several degrees hotter than present.
The IPPC models are wrong, on the conservative side, but it was known that they were wrong on the conservative side. The "surprise" is only that the wishful thinking that making conservative models provides a sense of security, turns out to be completely stupid.
However, decades ago to the present some climate modelers worked on realistic models (sometimes the same modelers that work on IPPC models too, and pointed out the things missing that make them conservative), which have always been terrifying in terms of the risk indication (numerical models of complex systems and things that haven't happened yet, only inform risk, never actually predict what will happen).
So, I'm not arguing solar energy can now arrest or reverse the climate crisis, only that it could have easily do so in an economically feasible way if the costs of fossil were internalized (whereas your magma technology could not have done likewise), and, even today, a massive proliferation of solar technology (and adapting society to use solar energy efficiently) would mitigate the crisis, but that is a political problem that is less and less feasible as the world is disrupted more and more by climate change and derivative affects.
So yes, billions of people are likely to perish, and it is definitely murder by the West in both a collective sense of apathy and specific sense on the part of the denialist industry, but that is not a "solution" proposed by us environmentalists pointing it out, it's just largely inevitable at this point.
Certainly, if you could prove magma energy to work, or even be "worth a shot", some billions should be spent finding that out and then a few 10s of trillion building your system if you turn out to be right.
Likewise, some billions should be spent on solar energy to discover the same, and some 10s of trillions spent converting the world to solar (in both installation of solar energy, and converting infrastructure to efficiently use it, with more local production using local energy, removing the large energy costs of both transport and large transport infrastructure).
Neither of these scenarios are happening, and spending trillions on bailing out the banks (i.e. corruption) from the consequences of their own corruption, in combination with the costs of more bailouts of the system in general due to the pandemic, and costs bailing out the system due to the affects of climate change, will likely lead to the kinds of economic dislocations that make large scale global investments no longer possible.
But the reason no alternative to fossil is being developed in a serious way (a way that would actually reverse carbon emissions) is political, not technological nor economic (we have the technology, and, as the world is discovering, eating the costs of climate change head on is not "economic bravery and realism" but complete idiocy).
As a note, water is an excellent mover of heat via convection, and if heat conduction to rock was instant as you say, then it would efficiently re-heat any hot water that's down there.
You can also look at the physics and economics of heat storage for power. They exist (such as for solar thermal energy), but for relatively short periods of time, because the material volumes required are simply massive. You'll also note that things like molten salt are used that efficiently transfer heat by convection, and if you work out just having a big cube of rock heated from below, it's not an efficient system (which would be analogous to having a a large cube of rock below the surface heated from further below that).
As an aside, the solution to the intermittence of solar energy is to simply match energy consumption to the energy availability as much as possible, which brings the problem down to a manageable level.