The OED doesn’t give much space to this, but it appears to me (at first look) to fit right in with this usage:
With adv. the: By that amount, to that extent, just so much.
a1616 Shakespeare As you like It (1623) i. ii. 92 All the better: we shal be the more Marketable.
1879 Tennyson Lover’s T. 82 He was all the more resolv’d to go.
In the OP there is a slight difference in sense that is one can probably attribute entirely to the rest of the construction: “that I can go”, so that the overall meaning is “as fast or far as we can go”. Overall, I think this is an organic development of the usage illustrated in the Shakespeare and Tennyson quotes, and one needn’t invoke German immigrants to explain it. I don’t recall any parallel construction in German that could have been calqued to form this, though my German is pretty limited so I wouldn’t rule out its existence either. Still, I don’t think that explanation is parsimonious.