Adler, Bobrowski and Weinberber's "Crackle: The Persistent Homology of Noise" is an answer to my question. I have not read it closely yet but it appears to confirm the guess in the question, and provide answers for other distributions as well.
Although this paper does not target my question directly it gives a more quantitative answer to a nearby question, that of the length of the largest barcode for certain types of random point clouds. Maximally persistent cycles in random geometric complexes.