Somehow "proving that there exists" seems a lot harder than "there is exactly the same number of roots and critical points". The second statement seems to match so well with the classical complex analysis toolbox...

Interesting! So I guess the desired inequality becomes $|z g'(z)| < | g(z) + z g'(z)| + |g(z)|$ on the boundary. Dividing by $|g(z)|$ (which never vanishes), we get $|z g'(z)/g(z)| < |1 + z g'(z)/g(z)| + 1$. This is true unless $z g'(z)/g(z)$ is on a real number $\leq -1$. If $z g'(z)/g(z) = -1$, we would have found a critical point of $z g(z)$ on the boundary, so it suffices to exclude $z g'(z)/g(z)$ being real and $<-1$...

Right, it's not just Rouche but it looks somewhat `friendlier' and similar to other problems I have seen solved via a combination of the Schwarz Lemma, Riemann mapping, Rouche, the Argument Principle, Koebe, ....

Ah, this is fantastic, this leads to a complete characterization! This is tremendously helpful -- thank you! I would award the bounty if it was an answer, I am not sure I can award it to a comment.

I don't think it's always possible. If we take (-1 + x) (-0.8 + x) (0.8 + x) (1 + x), then no matter what constant we choose for the antiderivative, the antiderivative cannot have more than 3 real roots. The problem is not only about the roots being distinct, they should also all be real.

I do see a somewhat similar curve for the adjacency matrix but p has to be quite small depending on $n$. Once $p$ is too large, all the eigenvectors are very nicely delocalized and the curve is quite flat. For the Laplacian $D-A$, it seems that we see the phenomenon even when $p=0.5$ and $n$ becomes large.

The curve suggests that the smallest values are on the edge. I had a look at the smallest nontrivial eigenvector and it does seem to localize in the vertex with the fewest neighbors. (I don't know what happens if there is more than one such vertex). The question is then somehow whether it localizes quickly enough to have a small $\ell^1$ norm. Some basic numbers seem to suggest that $\| v_2\|_{\ell^1} / \sqrt{n}$ is decaying as $n$ increases. I get $\sim 0.34$ for $n=500$ and $\sim 0.3$ for $n=5000$.