Why ordinary least squares degrades as p approaches n

Let $y = X\beta^\star+\varepsilon$ with $\varepsilon\sim\Ncal(0,\sigma^2I_n)$ and $X$ having i.i.d. $\Ncal(0,1)$ entries, $p<n$. Show that the expected squared estimation error of ordinary least squares is

$$\E\|\hat\beta-\beta^\star\|_2^2 = \sigma^2\,\E\big[\operatorname{tr}\big((X^\top X)^{-1}\big)\big],$$

and use the Marchenko–Pastur law to argue that this behaves like $\frac{\sigma^2 p}{n-p}$ for large $n,p$. Evaluate at $p/n = 0.5$ and $p/n = 0.95$.