Background

I read (what I could understand of) this article in which authors Yuan and Steinle-Neumann "...use advanced quantum mechanical simulations on silicate and metallic melts showing that hydrogen becomes increasingly more incorporated in metal over silicate at high pressure and temperature, conditions under which Earth's core formed. Therefore, hydrogen can be present in the core in high abundance. [...] The Earth's early accretion events, in particular magma ocean formation, had a great impact on the chemical and thermal evolution of the Earth. Segregation of metal from silicate to form a core during the magma ocean stage has removed elements besides iron and nickel from the mantle[...]. Light elements such as H, C, O, and Si enter the core as a consequence of this process." The article goes on to present a model for how light elements near the planet's surface could end up being transported to the core, leaving a mostly silicate crust with both light elements and metals mostly transported down to the core: light elements form weak bonds with iron under high pressures and are gravitationally sorted with the iron. "We find that hydrogen is weakly siderophile at low pressure (20 GPa and 2,500 K), and becomes much more strongly so with pressure, suggesting that hydrogen is transported to the core in a significant amount during core segregation and is stable there."

Question

How did diffuse hydrogen in the near-vacuum of the planetary accretion disk find itself bound, chemically or gravitationally, to the forming Earth in the first place?

As I understand it, the mechanism proposed for incorporation into the core necessitates extreme pressures and temperatures - which is good for explaining how it got from deep in the molten surface down to the core, but doesn't seem to fit planet formation in the first place. Why didn't the hydrogen swept out by proto-Earth simply boil off of the upper atmosphere as fast as it came in, never finding itself in a pressure regime sufficient to become chemically bound to iron?

It seems to me that intermediate chemical bonding of hydrogen into a less volatile form could explain it. I've read elsewhere that very little of Earth's water was swept out during accretion, but perhaps silanes and alkanes or other hydrogen-rich compounds might have formed in the accretion disk, been swept out, and then been broken up and re-bound with other chemicals in the extreme environment of the accreting planet. Or there might have been enough free Si, C, and other chemicals with an affinity for hydrogen in the atmosphere or surface of proto-Earth to bond with hydrogen, and then break up when convected down into higher temperature and pressure regimes. My chemistry knowledge is pretty weak, so I don't know if either explanation is plausible.

But let us contain a key result here, that the ratio of hydrogen to oxygen of course would be very different now to the protosolar nebula, i.e. $[12/100]\sim1$%, where this low number compared to the protosolar value that you might have expected, is coming from the assumption of initial water content of the rock.
Note that I am comparing atom numbers here, where their 1% is a weight-percent.

If you are interested, I can add a bit of discussion on how volatiles might get to the planet in the first place during the accretion process, but that would happen in the next few days.