I'm struggling to see how this makes sense. What's the evidence that someone can study advanced maths without understanding middle school algebra? It underlies calculus, analytic geometry, even a lot of combinatorics.

Analytical geometry underlies calculus,not Algebra.And combinatorics is a completely different arm of the two cultures of mathematics.Also Geometry was fine before Rene Descartes messed it up with Algebra and then we christened it Analytical Geometry.The method of exhaustion,amply developed by Archimedes with zero Algebra is the basis of intergration.It is because we start out with differentiation that we think Algebra is super important.Tom Apostol comes close in his calculus textbook when he actually starts with intergration.

A compromise would be to have two streams:The left-brained folks should follow the Algebra ->Geometry->Calculus track;the right-brained folks should have a Combinatorics ->Geometry ->Calculus track.

Hell, I think you need basic algebra for basic abstract algebra! And you're not convincing me you know anything about upper-level math if you don't know basic like... group theory.