The proof is not 'that they are spies', but could be any
document or a something very material and not abstract.
True,
but that's not the point.
that is not a pronoun in
the proof that they are spies. That is,
that does not substitute in any way for the noun
proof.
Contrast:
In the proof that the scientist provided, the scientist provided "that", namely, provided the proof.
In the proof that they are spies, we can't say, they are spies "that", namely, they are spies proof.
________
This is the confirmation that that they are spies is a content clause, that is, an appositive clause.
CJ