Finally! The method of proof was discussed in this post. Actually, the procedure for finding the sub groupoid is better than what I wrote in that blog post, and follows more closely the material in Keel-Mori.

We will soon add another lemma with more hypotheses where the output is a scheme, and not an algebraic space (namely when s, t are separated, locally of finite presentation, and flat). This avoids using Hilbert schemes at the cost of leaving the category of schemes temporarily.

This is a splendid example of an application of the theory of algebraic spaces: Namely, you define some functor, show it is an algebraic space, and then a posteriori you prove it is a scheme by some additional arguments.