Is shuffling a set of items after popping an item meaningfully more random than doing it once, before starting?
Say we have $n$ people. After shuffling the whole list (once), every person occupies every position with probability $1/n$.

Shuffling before each draw means every choice is uniformly random; that is, because the choices are now stochastically independent, being drawn as $i$th person has probability

$\qquad\displaystyle \frac{1}{n-i+1} \cdot \prod_{k=1}^{i-1} \Bigl(1-\frac{1}{n-k+1}\Bigr) \quad=\quad \frac{1}{n}$.

So no, shuffling repeatedly does not "add" any randomness -- whatever that would mean. It's uniform, i.e. "fair", to begin with, so what would "more random" mean?

The correct statement is: repeated shuffling does not change the distribution. Which means it is unnecessary, but also not harmful if it eases the mind of laymen participants.