If you want to dig into the specifics of any given model, well, you'd have to dig into the specifics of the model.
If you look at the Timeline of the Far Future, somewhere between 100m and 800m years from now life as we know it on Earth becomes untenable (asteroid impacts, solar flux), if not before (other effects).
There are some practical upper long-term limits to human population on Earth. I suspect it's anywhere between a few hundred millions to low billions (I don't believe present populations are long-term sustainable).
There are other resource constraints which may serve to set an upper limit on energy consumption per capita. From this we can establish long-term energy demands, available supplies, and feasibility of these supplies.
4 billion year forecasting is rather more fantasy than science, so far as human endeavors go.
If the model is physics, the specifics say that there's no such thing as renewable energy. Solar energy doesn't "renew itself", it's generated by the fusion of light elements in the sun. Geothermal energy doesn't renew itself, it's generated by the radioactive decay of heavy elements in the earth.
The machines we use to harness energy don't last forever either. If every 30 years I have to replace either 2,600 tons of equipment for energy source A or 200 tons for energy source B, don't I want B?
Yes, we all believe in the Third Law of Thermdynamics, and understand that the arrow of time is defined by entropy.
"Renewable" means that the specifically tapped resource will renew, by its own access to its prime driver, faster than we can deplete it. In your geothermal instance, it's the geothermal reservoir that's considered renewable. It's also possible to tap geothermal as a nonrenewable resource (generally in open-cycle systems in which groundwater is vented, release, or otherwise not returned to the reservoir, resulting in eventual depletion).
Even fossil fuels are renewable if the extraction rate is kept below the creation rate. Given that FF were created over roughly 200 million years, and will likely be practically exhausted in fewer than 200, that rate is on the order of 1 million times less than we're utilizing them now (probably more if you consider that extraction rates have accelerated the past 100 years).
Your harnessing plant isn't considered part of the fuel resource, and isn't considered in the sustainability equation, generally, unless you're doing something silly like open-pit nuclear fission to capture waste heat for generation, in which you'll likely exhaust your available habitable planetary surface.
Most likely you're not physically transforming your plant in a way that excludes, say, recycling of raw materials, and/or your plant is comprised of very abundant materials (iron, silicates, etc.) relative to your fuel source. It's not the absolute magnitude of plant material involved in generating energy, it's the portion of same relative to the available resources for replenishing these. Generally, structural materials are much more abundant than energy-containing materials. Starting with the matter surrounding and below you at this moment.
You act as if the distinction makes sense. Structural materials are more abundant than energy-containing materials? Hydrogen is the most abundant element in the universe. Structures, regardless of what they're made of, contain embodied energy by virtue of having structure.
Please point me at the large available free hydrogen reservoir on Earth.
Sources of useful energy are not hugely abundant. And, in the case of nonrenewable energy sources, are consumed in the process of utilizing them.
If you're looking at the Earth's crust, it's 60% silica, with most of the remaining 40% being various highly oxidized compounds, including certain quantities of dihydrogen oxide.
Over what period of time (or is the rate of consumption fixed?) and what region of space?
"at projected rates of consumption, would be sufficient for forseable demand until the end of time on Earth"
Not sure I believe 4B-year forward demand projections...
Isn't the amount of energy expended (and/or mass of raw materials mixed) per TWh produced what we really care about?