If I haven't miscalculated, such intersections (when smooth) are exactly the genus $4$ K3 surfaces. So probably potential density is expected but unproven except in special cases (covered by Bogomolov-Tschinkel).

I think the bound still applies - your objection implies that a stronger argument for a $(\ell(w)-\ell(u))!$ bound fails, but my bound only needs to fix an ordering for $w$ and not a distinguished subword corresponding to $u$.

A quick note: The same argument that you use in the $\ell(u)=0$ case gives a bound of $\ell(w)(\ell(w)-1)\cdots(\ell(u)+1)$ in general.

I believe this follows (in a general model category) from Proposition A.2.3.1 in Higher Topos Theory - the statement only covers the case where $Y$ is the final object, but the more general case follows via working in the overcategory over $Y$.

I think arxiv.org/abs/hep-th/0312282v2 contains something like what you want, but for $N=n.$ Unfortunately I wasn't able to find anything more general/precise by looking through papers that reference it.

Could you give some motivation for why you believe that the radical should be generated in the degrees you list?

Your argument seems correct, but I don't think it is that surprising that there is a simple construction. Looking at google scholar citations, it appears the article of Danzer-Grunbaum didn't receive much attention until the 1980s (maybe because Erdos popularized such problems around then?)

I think the essential point of the problem is exactly to prove that this subvariety is not contained in a linear subspace, which is not at all immediately clear from the statement. One can either interpret it as the image of the Veronese embedding, after which it becomes immediately clear (it's amazing the psychological effect dualizing has) or do something explicit as in mathoverflow.net/questions/98714/….