Why is it that the bigger the side-chain is, its hydrogen atoms tend to be more equatorial?
It's not exactly clear from your question what distances you consider, but I have taken snapshots of a methylcyclohexane with the methyl group in the equatorial and axial positions:

      (equatorial)

    (axial)

The distance of the methyl $\ce{H}$ atoms to the nearest cyclohexane $\ce{H}$ atoms are displayed, and you'll see that indeed the axial position has a shorted $\ce{H–H}$ distance of ~ 2 Å, while equatorial position has longer $\ce{H–H}$ distances.

Also, as others have remarked in the comments: only the groups (and $\ce{H}$ atoms) directly on the cyclohexane ring are said to be in axial or equatorial position. Atoms of side chains do not follow this terminology, in particular for side chains that can rotate freely as it means nothing there.