A rough sketch.

Each polynomial $P_n(x)$ with positive coefficients $a_i>0$, for $0\leq i\leq n$ induces a probability distribution on the set $\{0,1,\dots,n\}$ with probability weights specified by the coefficients of $P_n(x)$ Namely, a random variable $X$ is distributed according to $P_n(x)$, if, for $i=0,1,\dots,n$, $$\text{Prob}(X = i)=\frac{a_i}{P_n(1)}.$$ If $X$ and $Y$ are random variables distributed according to $P(x)$ and $Q(x)$, respectively, then $X + Y$ is distributed according to the product $P(x)Q(x)$. In particular, let $X_1,\dots,X_r$ be independent random variables distributed according to $P(x)$. Then their sum $X_1+\cdots+X_r$ is distributed according to $P(x)^r$. The central limit theorem therefore suggests that the coefficients of $P(x)^r$ should eventually become “more log-concave” as $r$ increases.

We know this: if a log-concave polynomial has no internal zeroes then it is unimodal. This applies to your polynomials $P_n(x)$ here, due to the hypothesis $a_i>0$. Hence, your polynomials $P_n(x)^r$ will eventually settle down to unimodality.

Caveat. Single peaks are not guaranteed by this argument.

A rough sketch.

Each polynomial $P_n(x)$ with positive coefficients $a_i>0$, for $0\leq i\leq n$ induces a probability distribution on the set $\{0,1,\dots,n\}$ with probability weights specified by the coefficients of $P_n(x)$ Namely, a random variable $X$ is distributed according to $P_n(x)$, if, for $i=0,1,\dots,n$, $$\text{Prob}(X = i)=\frac{a_i}{P_n(1)}.$$ If $X$ and $Y$ are random variables distributed according to $P(x)$ and $Q(x)$, respectively, then $X + Y$ is distributed according to the product $P(x)Q(x)$. In particular, let $X_1,\dots,X_r$ be independent random variables distributed according to $P(x)$. Then their sum $X_1+\cdots+X_r$ is distributed according to $P(x)^r$. The central limit theorem therefore suggests that the coefficients of $P(x)^r$ should eventually become “more log-concave” as $r$ increases.

We know this: if a log-concave polynomial has no internal zeroes then it is unimodal. This applies to your polynomials $P_n(x)$ here, due to the hypothesis $a_i>0$. Hence, your polynomials $P_n(x)^r$ will eventually settle down to unimodality.