What is the consequence if there are only polynomially many 'yes' classes of instances of a language that is polynomial time reducible from a problem equivalent to UnambiguousSAT (such as possibly unique subset sum)?
It puts NP into P/poly, and therefore collapses PH to its second level.
By basically the same as the usual proof that BPP is in P/poly, there is polynomial advice that provides good random bits for the randomized reduction of Valiant-Vazirani. Use that advice to produce the queries to UnambiguousSAT. Apply the reduction from UnambiguousSAT to the sparse language $S$ - which is in P/poly - and use the additional advice to decide $S$.