> Neither [degrees nor radians] is really wrong. Degrees are strange, ...

That sounds like the pot calling the kettle black. 'Irrational' means ... errm ... strange. 'Transcendental' was the word Leibniz reached for when he realised he had something stranger than irrationals.

> an artifact of the Babylonian system with no real mathematical significance.

Hmm? An artefact of being approximately the number of days in a year (which is as true for us as the Babylonians), and a number which has many factors, so can happily measure quarter-turns, eighth turns (bracing to keep your right-angles upright), whole-number of degrees for internal angles of all the faces needed for the Platonic solids ...

I would have thought there's a teachable moment there about prime numbers, factorisation, and for bringing fractions into a coherent continuum.

It's not like ignorance of this alleged "mathematical significance" of radians prevented building Hanging Gardens or Pyramids, Stonehenge, the fractals of the Walls of Benin, Machu Picchu trapeziums or aqueducts or anything. (A bit like Category Theory, really.)