With the recent achievement of extremely high-quality epitaxial interfaces between InAs nanowires and superconductingAl shells with strong superconductor-semiconductor tunnel coupling, a new regime of proximityinduced superconductivity in semiconductors can be explored where the induced gap may be similar in value to the bulk Al gap (large gap) with negligible subgap conductance (hard gap). We propose several experimentally relevant consequences of this large-gap strong-coupling regime for tunneling experiments, and we comment on the prospects of this regime for topological superconductivity. In particular, we showthat the advantages of having a strong spin-orbit coupling and a large spin g factor in the semiconductor nanowire may both be compromised in this strongly coupled limit, and somewhat weaker interface tunneling may be necessary for achieving optimal proximity superconductivity in the semiconductor nanowire. We derive a minimal, generic theory for the strongcoupling hard-gap regime obtaining good qualitative agreement with the experiment and pointing out future directions for further progress toward Majorana nanowires in hybrid semiconductor-superconductor structures.