The first four cites (out of six) in the OED for “high-street” as an adjective all use “High Street”, no hyphen (and, indeed, initial caps).

If a phrase has caps, it doesn’t need a hyphen; in general, there’s a rule of parsimony about these things.  (The hyphen is needed to make clear that the two previous words are to be taken together, as a unit; if they have caps, that’s clear already.) And you don’t know what majority usage is unless you do a corpus study, which presumably the OED editors have.

Strangely, until the 1950s, UK newspaper style for “High Street” as a noun, took the form “High-street”, as it did for other streets and roads: “A girl was killed and two men were injured in a collision at Ealing yesterday between two motor-cars at the junction of Uxbridge-road and Gunnersbury-avenue.”

I had no idea that style was in use so recently. It was normal in 18th-century Britain.