I think a cycloid as the tangential velocity will vary between 0 and double the initial, and the perpendicular velocity will vary between plus and minus the initial at the same frequency.

I think these are the parametric equations

X(t) = r * omega * t - r * cos(omega * t)
Y(t) = r sin(omega * t)

From a physics A-Level perspective I have come to teach it that way to preempt the textbooks - I'd rather get in first and make sure they're using it right than leave it to chance when they come across it.

If I was writing a textbook I think I'm with you and I'd stop at centripetal acceleration.

I teach it in both Further Maths and Physics. I find that naming the force helps discussion, but I make sure they're bored of me saying that Centripetal Force is a Resultant Force before we even get to the equation.

Draw force diagrams and get them to label the resultant force before naming it.