Solid carbon is basically coal. It has more usable energy per mass than any hydrocarbon, and just a little bit less per carbon.
And we can get this energy from sunlight.
But nobody is planning on deploying something like it currently anyway.
Take take the gallium example, for atmospheric levels of CO2, is 400:1. Add the thermodynamic minimum to reverse GalliumOxide to Gallium and O2, and its 800:1, at an absolute minimum. This crushes the difference to methane.
More importantly its a pointless argument, we have practically endless nuclear fuel available, enough to supply the world with power entirely for millions of years. So it does not matter if its inefficient, its climate neutral anyways as long as we stop being idiots and use what we have better. That said there are way simpler more cost effective methods than using a rare earth metal.
To make it concrete, let's look at methane, the hydrocarbon with most hydrogen.
One kg of methane contains 0.75 kg of C and 0.25 kg of H. When it burns it realeases 55.6 MJ (per wikipedia [1]). Carbon's energy density is 32.7 MJ/kg, so the 750 g release 24.6 MJ. More importantly, you could in principle split the CO2 for the cost of 24.6 MJ and be left with a net of 31 MJ. Q.E.D.
Well, except in real life nothing is done with 100% efficiency. You can't just use all the 55.6 MJ released by burning CH4, you first convert it to electricity, and the best you get is 63%, so you make 35 MJ of electricity. If you get 70% efficiency in splitting CO2, you get exactly the 24.6 MJ you need. But you don't get that efficiency. But let's just say you get even more than that, let's say you get a whooping 85%. That means you need 30 MJ for the splitting, and you are left with a net of 5 MJ of electricity. Which is another way of saying you increased the cost of electricity you generate by a factor of 6 (=30/5), this ignoring altogether the capital cost associated with the splitting of CO2.
But all is not lost.
If we get back to 1kg CH4 = 0.75 kg C + 0.25 kg H2, burning the CH4 we get 55.6 MJ, burning the C and H2 separately, we get 24.6 MJ + 35.5 MJ = 60.1 MJ, which is 4.5 MJ more. That's a different way of saying you need 4.5 MJ to split CH4 into C and H2. That reaction is called methane pyrolysis [2]. Let's say you manage to deliver this with only 30% efficiency, i.e. for 15 MJ. You are left with 35.5-15=20.5 MJ net energy. If you convert this to electricity you get about 13 MJ, which is not that great. But hydrogen is valuable in itself. If we move towards the hydrogen economy, this method of generating hydrogen may be the winner.
[1] https://en.wikipedia.org/wiki/Energy_density
[2] https://en.wikipedia.org/wiki/Pyrolysis#Methane_pyrolysis_fo...
Burning fossil fuels for energy: hydrocarbon + O2 -> atmospheric CO2 + other stuff
Sequestration: atmospheric CO2 -> not-amospheric-CO2 e.g. solid C + other stuff
Notice that those are not exact reverses of each other, so there is no requirement that the energy released in one be the same quantity as the energy absorbed in the other.
Hydrocarbon fuels have very high Gibbs free energy. Solid carbon does not. We're only trying to get the CO2 out of the atmosphere, not return it to the form of the original fuel that was burned. Therefore, sequestration can potentially require less energy than was originally released from burning the hydrocarbons.