This is of course entirely expected. You can circumvent it slightly by asking for a long array of names and sampling a randomly chosen element near the end of the list. Say ask for 50 names and use the 41-50th element stochastically.

Not perfect, more expensive, but it helps a little. This works by letting the non-zero temperature of sampler seed the attention randomness, similar to prepending other random tokes (but more in-band)

Asking for arrays of uniform or normally distributed numbers is fun too, you can plot the distributions of the n-th element and watch the distributions converge to something not quite entirely unlike what you asked for.

Often there's some bias between element indices too, eg. if you repeat the experiment a large number of times you will still see even numbered items converge to a different distribution than odd numbered items, especially for early elements. Hence the stochastic averaging trick over the last few elements.