The thing is even Hawking is saying that they picked a certain path based only on "modesty ".
Regardless of what (you think) Hawking said, the fact is that this isn't true. There are really three key figures in this: Copernicus, Tycho, and Kepler. The first is important as the originator of the idea, but it is the third who is the most important figure in cementing the heliocentric model as superior, because he's the one who worked out elliptical orbits. You have to understand how the Ptolemaic model worked to appreciate the development. Ptolemy, being a man of his time, insisted that the motion had to be circular; but astronomers could already see that simple circles around the earth didn't give accurate predictions of position (especially the phenomenon of retrograde motion). So they imposed secondary circles called epicycles, so that (for instance) Mars orbited around around a point on Mars's primary circular orbit. This got them closer, but it was still imprecise, and led to adding more epicycles on the epicycles. So along came Tycho, who was a very good astronomer with a commitment to the Ptolemaic model. And he started messing with the epicycles, and eventually heads off to a model where the primary epicycle for each planet is a duplicate of the sun's principle orbit, with each planet's location on those epicycles in correspondence to the sun's position. In other words, he's simply taken the Copernican model and put the fixed point at the earth instead of the sun.
That is really the only point at which there is still something of an arbitrary choice between the two models, because Kepler's observation gets rid of the epicycles and the circular motion which were the chief cause of the difficulty. He instead not only worked out elliptical orbits, but came up with a very simple formula for the motion along those orbits. Each such ellipse has two foci, and the body being orbited sits at the one of those foci. For the moon, that point is where the earth rests; for the planets, it is where the sun sits. You can of course make a hybrid model like Tycho's final version, but it's a model in which only the sun and the moon are geocentric, and all the other planets are heliocentric; in any case this model was so much more accurate and so much simpler that it killed the Ptolemaic model completely. The discovery of the Galilean moons of Jupiter pushed this further along since it was obvious that whatever model you came up with had to have them effectively orbiting Jupiter; a contrived orbit of earth was outlandishly complicated in comparison.
Newton's theories of gravitation and motion explained why Kepler's laws worked, as well as getting rid of almost all the rest of the error (there's another small correction arising from relativistic effects). You cannot make geocentrism work in Newton's system; you have to come with an extremely contrived mess in which the earth's motion, and only
that motion, follows a different set of rules. The math quickly becomes too complicated to take seriously, so nobody bothers; it's a lot easier to work from the observation that things work just fine if you assume that the earth follows the same laws as everything else. Throwing the relativistic effects into the mix only make it that much worse.
It's not just that a geocentric model is more elegant; it's that, in the solar system observed in isolation, a relativistic model based on Newtonian mechanics is far and away the simplest, and that alternate versions either don't work (e.g. the Ptolemaic system) or are simply Newtonian models with an exception for the earth. All the stuff Hawking is talking about is at a larger scale, galactic in the case of dark matter and intergalactic in the case of dark energy. They are being introduced because the apparent deviation from Newtonian/relativistic motion is most readily explained by keeping the model as is and introducing unseen objects and substances which act within that model to produce the observed deviations. Personally I think dark matter has a better chance of long-term survival as a real phenomenon, but in any case the whole thing is still something of a theoretical kluge that wants more verification. The expansion of the universe is a phenomenon which also needs explanation, but again things are pretty speculative; the notion that we have some idea of what set off the big bang can be rejected out of hand as the purest of speculation.