 Solution

Run Wilson algorithm with root $a$, beginning with a simple random walk from $x_1$. Its chronological loop erasure is added as the first branch, so the resulting tree contains a path from $x_1$ to $a$ with exactly the law of the stated loop-erased random walk.

For completeness, the algorithm produces a uniform spanning tree. For any prescribed ordered collection of loop-erased branches, the random-walk decomposition gives a product of local transition factors and diagonal killed Green functions. The transition factors depend only on the resulting oriented tree, while the product of Green functions is independent of the vertex order by Question 3. Thus every spanning tree receives the same probability. Since a tree has a unique simple path between two vertices, its $x_1$-to-$a$ path has the claimed law.