Sure the price of certain forms of energy (wood fire, oil ...) may go up over time but while that happens cheaper sources appear and ultimately take over.
What's apparent to me is that we live in a unique era where energy is abundant and cheap, which may never be repeated (unless fusion power is developed).
(You can check my math yourself: total world energy consumption = 109,136 TWh [1] / 8760 hours/year = 12.45 TW / 15 W/foot solar panel output [2] = 830B square feet / 27878400 ft^2/mi^2 = 29772 mi^2 = 150x200 mi array.)
Working through his math, I think the problem is that the UK is unique in a.) having a high population density b.) with a large per capita energy consumption and c.) a lot of cloud cover. On his chart, only Japan, South Korea, the Netherlands, and Bahrain had it worse. The U.S. and China both had an order-of-magnitude better energy consumption per unit area; most of the world had two orders of magnitude. That's an argument for why Britain is fucked in the new globally-warmed, fossil-fuel-limited world, but not why the world is fucked. A lot of developing areas could see their energy consumption rise to rival the U.S. and still not bat an eye with solar power.
The EROEI of solar panels is between 8.7 - 34.2; wind is 20-50. [3] As long as this number is > 1 (which it is for nearly every energy source except biofuels) you can keep building production capacity indefinitely; you just invest the energy from previous capacity into building new capacity. Higher numbers of course let you do more other stuff while this is going on and build out faster, but it's just like capitalism. As long as you turn a profit on each panel, you'll have more as time goes on.
[1] https://en.wikipedia.org/wiki/World_energy_consumption
[2] https://www.solarpowerrocks.com/solar-basics/how-much-electr...
[3] https://en.wikipedia.org/wiki/Energy_returned_on_energy_inve...
Yes, the UK does have issues by these metrics, and the US is in the unique position of a huge per capita energy consumption and huge land reserves. Therefore, it is somewhat feasible in the US to just keep building wind and solar farms. However, I wonder how the economics of the non-renewable minerals needed would affect the price. It also doesn't (as far as I know) take into account all the extra infrastructure required to support the fluctuating nature of renewable energy sources.
I thought about the rare-earth issue, but there's a simple solution that bypasses both it and the energy-storage issue: use solar-thermal plants instead of photovoltaics. These require only mirrors and steam turbines, both made out of abundant materials. Efficiency is actually higher than photovoltaics (30% or so), and energy can be stored as heat for up to 5 hours before it's used. (Ironically, they have the same issues relative to photovoltaics that photovoltaics have over fossil fuels; in theory they're simpler and more environmentally friendly, but implementing the theory correctly requires solving a number of engineering problems that together make them cost more and generate more hiccups than photovoltaics have.)
The higher cost isn't actually true anymore, for photovoltaics and wind at least. Solar hit grid parity with fossil fuels in late 2015, and is now actually cheaper than coal power plants. This (rather than subsidies) is why the coal industry is in deep shit: it just doesn't make economic sense anymore, and the electric industry is rapidly switching over to renewables & natural gas.
This is not at all obvious to this casual observer. Earth radiates heat away at a constant rate. We can never produce more energy than the earth radiates away, for obvious reason. With current trajectories we will release more energy than earth radiates away in less than 200 years. I'm not sure how the price of a limited resource can go to zero, but I'm also not very well versed in economics.
In practice, the total energy consumption of the Earth is bounded by the amount of solar energy that falls on the earth, which is about 100,000 times greater than current total world energy consumption. For what civilizations might look like that exceed that, check out the Kardashev scale [2], which classifies civilizations by the size of the astronomical object whose energy they harness. Type 1 civilizations harness a planet, Type 2 harness a star, Type 3 harness a galaxy. We're a lowly Type 0 civilization, which means that we have not yet attained the technology needed to make full use out of all of our planet's resources.
[1] https://en.wikipedia.org/wiki/Black_body#Radiative_cooling
The temperature in the formula is given in Kelvins, so for all practical purposes (300K less than 10°K variation), the 4th power means every degree increase the amount of energy radiated by ~1.3%