Possibly some confusion here? The first all line timetable, in 1974, has daytime trains to/from Leamington Spa using Moor Street but evening services using New Street.
I think it was around 1969 maybe 1970 that the Leamington trains were routed into New Street.
It didn't last for very long and I suspect for political reasons to help the case for closing Moor Street. Once Moor Street was reprieved, most trains reverted to Moor Street as the terminus and once services were improved by WMPTE Moor Street was no longer an option, rather a necessity.
The down side of course, is that connections to the north from places like Warwick and Solihull do involve a bit of a walk across Birmingham.
I can never understand why XC don't stop at either place. A Solihull stop for New Street could be highly useful as well as bringing a through train to further afield.
I suspect XC aren't interested in developing the Solihull route because they would rather be running via International but that's been the case for too long now and still seems just as unlikely.