I haven't been following this, but just based on a few hours of eavesdropping--there's perhaps a category error in the argument's reading of history. The usual question is whether the equitable remedy existed in UK equity practice at the founding. But the question of whether a claim sounds in law or equity is distinct from this, and requires some imagination, since the two have been unified in the interval.
A claim sounding in law (or constitutional law) can have an equitable remedy, possibly binding those involved in the action; a claim sounding in equity at the time of the claim (e.g., one seeking a universal injunction, without statutory basis, as the ultimate remedy) arguably should be subject to the traditional common-law limitations on equity jurisdiction.