Tim, Ringo's vocals with Rory's band were limited in those days (as they were with the Beatles), so perhaps the songs you cite are the only leads he had pursued. Starr Time was more a showcase for Ringo's drum solos than it was for his singing.
Interesting trivia: A year after Ringo left the Hurricanes, they released their first single: "Dr. Feelgood," which no one on the planet bought, except Rory's Mom. So in 1964, Brian Epstein of all people would produce the next single for Rory Storm and the Hurricanes. No one bought that either. The band started to literally die off shortly thereafter:
Guitarist Tyrone Brian in 1966 from complications during route appendix surgery.
Rory (real name: Alan Caldwell) in 1972 from a drug and booze overdose.
John Byrne (also known as Johnny Guitar) died in 1999, perhaps from illness or advanced age.
Only two ORIGINAL Hurricanes remain -- drummer Ringo and bassist Lewis Walters. (Lewis actually had left the band before "Dr. Feelgood" was released.) Coincidentally enough, only two original Beatles remain: drummer Ringo and bassist Paul.