rolling dice - win if the sum of rolls is exactly $n$

As you remark, for $n>6$ the probability that you land on $n$ is the average of the $6$ predecessor probabilities. As such, it can never exceed the maximum of those $6$.

It is clear that $n=6$ has the greatest probability of the first six values (easy to check this by hand, of course). Thus no value beyond $6$ can be more likely, as iterated averages must lower the maximum. Thus the maximum value occurs for $n=6$, for which the probability is just a little greater than $.36$

It's not even close. $P(5)\approx .30877$ and $P(n)<.3$ for all $n\neq 5,6$. The limiting value, for large $n$ is $\frac 1{3.5}\approx .28571429$ since the average toss of the die is $3.5$ This limit is reached fairly quickly, as $P(26)\approx .28574$