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So what do you make of the go in my example He will go to the house? It's clearly in the same position as, e.g., the to do in He is going to do his homework, another perphrastic future, this time in go.



No, it is not in the same position. Will and shall are auxiliary verbs for the future tense, also known as modal verbs. There is absolutely nothing periphrastic about either one.



That's just what periphrastic means in this context. A non-periphrastic (or inflected) future construction would be one where the verb was used with some future marker built in. Spanish, for example, has both periphrastic and inflected futures, e.g., va a hacer (he is going to do, where hacer is the infinitive) and hará. We don't have the inflected future in English. If you don't like "going to do", how about "he wants to do his homework"? This would be a periphrastic construction of mood, rather than tense, but the principle is the same. The German would be er möchte seine Schularbeiten machen, where machen is the infinitive.

If you want to call one form of the infinitive the base form and reserve the term infinitive for the infinitive preceded by the preposition to, that's OK by me, I guess. I'll stick to calling them both infinitive.