We construct a family of translationally invariant lattice models with a large number (N) of orbitals at every site coupled together via single-electron tunneling. By tuning the relative strength of the electronic bandwidth and on-site interactions, which have a modified Sachdev-Ye-Kitaev form, we demonstrate a number of unusual features at strong coupling and in the large-N limit. We find examples of (i) an intrinsic non-BCS superconducting instability arising out of an incoherent non-Fermi-liquid metal, and (ii) an instability of an incipient heavy Fermi-liquid metal to superconductivity with transition temperatures comparable to its renormalized bandwidth. At strong coupling, these solvable models display pairing instabilities that are not driven by any special "nesting"properties associated with an underlying Fermi surface. © 2020 authors. Published by the American Physical Society.