The world has moved on since then, both immigration and train services, and it is no longer 'perfectly feasible' with today's requirements. As has been explained ad nauseum in this section previously.My personal thoughts on joining Schengen (yes, absolutely) aside, it's not actually necessary to fixing the Eurostar mess. Instead, we simply need to return to on-train passport checks, which are perfectly feasible—they were routinely done on the continent pre-Schengen, and indeed on Eurostar in the early days of its operation.
(Of course, the current government won't do this, as juxtaposed controls are a means for the UK to skirt its legal obligations regarding refugees.)