I've verified the statement for all $k\leq 13$ with the following greedy algorithm combined with (multiple) initial random shufflings.
Greedy algorithm. Given an ordered list with $k$ values of $x$'s, we start with $s=0$, and at each of $k$ steps we extract and add to $s$ the first element $v$ from the list such that $0\leq s+v<1$. The algorithm succeeds if we are able to extract all $k$ elements from the list, and fails otherwise (when at some step a suitable element does not exist).
From Fedor's observation it follows that for a fixed $k$, we can restrict our attention to values $x_1 = \frac{m}{2^k-1}$ for $m\equiv 2\pmod4$ in the interval $[2,2^k-2]$ and the sequences of $x_i$ emerging from them. For each such sequence, we try the greedy algorithm on its various random shufflings until one succeeds.
PS. The order of nonincreasing absolute values suggested by Alex Ravsky above works well for a noticeable fraction of sequences and can be used before going into random orders, although this fraction decreases as $k$ grows.