The use of interbreeding to define the boundaries of species is convenient but does have its problems.
For example, you can have a breeding “cline” of, say, populations A, B, C, D, E, F, where organisms of adjacent groups can interbreed (A&B, B&C, C&D, etc.) but A & F cannot.
And failure to interbreed is not always genetic. Populations might be able to interbreed successfully in terms of genetics, but are prevented for reasons that are geographical, behavioral, or even anatomical.
I’m not clear how you argue that “God sets a genetic boundary” if, for example, He allows the genetics to work out ok but offers an anatomical impediment to interbreeding.
You’re right that there are some complexities to this but that is no reason to abandon a perfectly good and fairly common sense tool we can use to categorize creatures.
A good rule of thumb is that if two types of animals can breed (or if they can both independently breed with another species as you noted), that is positive evidence of a common ancestry, but the inability to breed can never in and of itself be positive evidence of a lack of common ancestry.