"Supersessionism would say, 'Yeah, but now those things are actually fulfilled in the Church,' and you have to make the case biblically, somehow...God kinda transferred those promises to the Church, which would seem to require something rather explicit."
Actually, He transferred them to His Son...and not really transferred, but they were originally
made to Him.
Now to Abraham and his seed were the promises made. He saith not, And to seeds, as of many; but as of one, And to thy seed, which is Christ. ... Wherefore then serveth the law? It was added because of transgressions, till the seed should come to whom the promise was made; ... For ye are all the children of God by faith in Christ Jesus. ... And if ye be Christ's, then are ye Abraham's seed, and heirs according to the promise. - Galatians 3:16, 19, 26, 29 KJV
I'm not sure how much more explicit it could be.