(Of course its easy to say that one should "do math" in lectures rather than just "tell math" but probably very difficult. Having been taught by Tom Korner, I found that he had almost, after many, many years of teaching, got this down: After stating some lemma, he would successfully put on the act of thinking through the proof on the spot by saying aloud the thoughts that any analyst would have when asked to prove the lemma and then turning those thoughts into the proof, in a way that seemed natural. (This also relates to fedja's comment elsewhere on this page))

Regarding a), I certainly agree that one of the major goals is to learn to do* mathematics, but regarding b) & c), I am led to remember Prof Tom Korner's essay "In Praise of Lectures". Roughly speaking, he argues that the real point of a lecture is for students to actually see mathematics be done . So a good lecturer actually does mathematics, rather than just lists what needs to be learnt. Thus e.g. the basketball example must be replaced by e.g reviewing videos of a basketball game, which can be instructive to someone who wants to play the sport well. (1/2)

I am not bothered about making any stipulations about multiplicity or normalization. I guess I could really ask it for a general linear combination $f = a_1\phi_1 + \dots + a_m\phi_m$; $a_j \in \mathbb{R}$.

Thanks for the comment fedja, I see your point. I don't know what made me think it was true now.

I know very little about the math going on in the background but a colleague demystified this for me by saying that basically it is a special feature of the basis you have chosen, namely the monomials. If you choose a different basis or even just weight each monomial by a factor, the roots will tend to congregate on a different set. Basically although a) any polynomial can arise and b) you chose them randomly... They aren't as generic as as you think; they've in fact been chosen in a special way.