I’ll focus on cosmological arguments (understood broadly to include design arguments) but much of what gets said will apply to the OA too. My claim is only that if the theistic argument is true, that these things follow.
1.) There is another meaning and mode of truth. Just as a geometrical or algebraic proof shows that there is another way in which things can be true than being manifest to sense and given in physical theory, a successful theistic proof shows that there is another way things can be true than being manifest to sense, given in physical theory, or a solution to an equation that is consistent with defined quantities.
2.) There is another mode of analyzing the sensible and/or finite. All analysis divides the complex into the simple, but this happens in different ways in math and physical science. In physical science, the desire is to reduce everything to what is most simple in the material order (the fundamental particles or principles) in mathematics the goal is to reduce everything to the simplicity of counting (one unit on another) or the simplicity of equality (where things have no real difference, only a different way of being understood) or the simplicity of the infinite (which is pure quantitative order and relation with no absolute first or last). A theistic proof analyzes the finite into another mode of simplicity , similar in some ways to mathematics and in other ways to physical science, but also differing from both.
3.) There is another sense of science. Science is nothing but the systematic order of truths, so this is a corollary from 1 and 2.
4.) There is another sense of “existent”. Proving that non-Euclidean quantities exist means something very different from proving that dark matter or phlogiston exists. We can’t very well argue that there must be dark matter because it is a possible solution to a defined physical reality that admits no manifest contradiction, but such a way of proceeding would suffice to prove the reality of, say, complex numbers.