The Sun is giving us a low entropy, not energy

First, some preliminaries: We always wish to have a system that can do useful work, e.g., run a water wheel, raise a weight, or generate electricity.

The catches are that energy is conserved (which you probably knew about) and also that entropy is paraconserved (which you might not have known about). Specifically, entropy can't be destroyed, but it is transferred when one object heats another, and it's also created whenever any process occurs, anywhere.

The problem with producing work arises because work doesn't transfer entropy, but heat transfer does (while also creating some entropy). Therefore, we can't simply turn thermal energy (such as the energy the Sun provides) into work; we must dump the accompanying entropy somewhere as well. This is why every heat engine requires not just a source of thermal energy (the so-called hot reservoir) but also a sink for entropy (the so-called cold reservoir).

In the idealized process, when we pull energy $E$ from the hot reservoir at temperature $T_\mathrm{hot}$, the unavoidable entropy transfer is $$S=\frac{E}{T_\mathrm{hot}}.$$

Now we extract some useful work $W$ (by boiling water and running a steam turbine, for example), and we dump all that entropy into the low-temperature reservoir at temperature $T_\mathrm{cold}$ (using a nearby cool river to condense the steam, for example): $$S=\frac{E-W}{T_\mathrm{cold}} .$$

The energy balance works out: $$E-W=(E-W).$$ The entropy balance works out: $$\frac{E}{T_\mathrm{hot}}=\frac{E-W}{T_\mathrm{cold}}.$$ The efficiency is $$\frac{W}{E}=1-\frac{T_\mathrm{cold}}{T_\mathrm{hot}}.$$ And the higher the temperature of the hot reservoir, the more work $W$ we can pull out while satisfying the two conversation laws.

Now to the point: The Sun sends a lot of energy our way: around 1000 W/m² at the earth's surface. But is this in fact all that much energy? The heat capacity of soil is about 1000 J/kg-°C, so if we simply extracted 1°C from a kilogram of soil per second, we'd match the Sun in energy per square meter. And there's a lot of soil available, and its absolute temperature is pretty high (about 283 above absolute zero in divisions of °C).

And the heat capacity of water is four times as high! Even better, water is self-circulating, so in this scenario, we could cool seawater and let it recirculate. We could operate a party boat: pull out thermal energy from water to make ice for our cocktails and use the extracted energy to cruise around all day.

Unfortunately, the restrictions described above tell us that we can't perform this extraction: there's no lower-temperature reservoir to send the entropy to (here, I'm assuming that most of the earth and atmosphere available to us is at around 10°C). In contrast, the Sun's temperature is enormous—around 5500°C, which makes the denominator of the effective entropy term $S=E/T$ relatively small. Thus, it's not the energy of the sunlight that's particularly useful—it's its low entropy.

A conceptual answer in two parts:

First, note that the energy of the Earth is essentially constant. The Earth continuously loses energy to space, and the Sun makes up that loss. (Yes, there are small plus and minuses, but this is basically correct) The Sun’s power is certainly not rapidly increasing Earth’s total energy.

So why does The Sun’s power seem so vital? Well, it makes up the lost power. Surrounding Earth in a Giant Space Comfort Blanket would reduce those losses too, but somehow that seems less great than the concentrated power of the Sun.

So that’s where entropy comes in: the Sun’s energy is concentrated & high temperature, hence low entropy (which is good), unlike diffuse & low temperature high entropy (less good) planetary heat.

Viewed that way, while just making up lost power, the Sun is providing a dose of order (low entropy) which allows life to do its thing by consuming that and giving off the power as disordered low-grade heat.

The entropy of the earth+sun system is lower than a system with the earth surrounded by diffuse energy equivalent to that of the sun. Technically, both systems have the same energy, but the former has much more usable energy.