Concentration bound for absolute value of sum of Bernoulli variables

You can try and see what these values should be under the Central Limit Theorem, and then use the Berry-Esseen bounds to prove that your guess is correct. You can't get large deviation bounds this way since the error is too big (and indeed, the normal approximation doesn't hold for the tails), but for your purposes it might be enough.

(I'm just a student so I'm sorry if I offend anyone with obvious comments)

If one takes the intuitive case $x_i = 1/\sqrt{n}$, the exact distribution is to linear transformations that of the Binomial distribution. Naturally, one should not expect to understand the tail bounds of your case better than this well-studied one. Additionally, if one is interested in "the worst case", then I phantom it should be precisely this. If not, perhaps the more successful approaches carry over.