Am I right?
As Kora's answer notes, yes.
What more can I add?
As you note, the variance of the sum is the sum of the covariances, not the average of the standard deviations. That seems sufficient, but you can certainly add more if you like.
For example, the thing I immediately noticed about the question was that you could know that it was false simply by unit analysis, even if you did not know that the variance of the sum was the sum of the covariances.
Suppose we have some Gaussian distribution that is representing some process measured in dollars or meters, or whatever. Dollars, since your example is financial. The standard deviation is also measured in dollars -- that is, how many dollars around the mean do we find most of the values in this distribution?
The variance is the square of the standard deviation, so its units are $2, whatever that means. So if we have:
As the number of assets in a portfolio increases, the variance of the portfolio returns will tend towards the average standard deviation of the assets of the portfolio.
The variance of each asset has units of $2, so the variance of the sum of assets does too. But the standard deviation has units of $, and therefore the average standard deviation does as well. It's not meaningful to say that a thing denominated in one unit "trends towards" a thing denominated in a different unit! Therefore we know that the answer has to be "false".
Similarly, if I said true or false, we've got a bunch of spheres in a box, and as we add more spheres by some process, the average surface area trends towards the average radius, you'd immediately know that had to be false even if you knew nothing about spheres. Area is measured in square meters and radii are measured in meters, and so one cannot trend towards the other.