 Solution

For $i=1,\dots ,n-1$, make one block from the edge weights $\{X_{ij}:j>i\}$. These $n-1$ blocks are independent because they contain disjoint edges. Changing one whole block only changes weights of edges incident with vertex $i$. Every matching contains at most one such edge, so its total weight changes by at most one. Comparing an old optimal matching in the new weights and a new optimal matching in the old weights proves that the maximum matching weight also changes by at most one.

Apply the bounded-differences martingale argument to these $n-1$ blocks, givingExposing each edge separately would give a weaker quadratic-in-$n$ denominator. The grouped exposure is what supplies the requested linear denominator. The proof needs only independent weights in $[0,1]$, not identical uniform laws. For $n=1$, the matching weight is deterministically zero; the divided formula is intended for $n\ge 2$.

The general grouping argument is vertex-block concentration of maximum matching weight.