This isn't a complete answer, but may help. An alternate line of attack is to look at the history and etimology of the word 'Valet' itself. I can't vouch for the reliability of the source, but etymonline.com drops these suggestive titbits:
Modern sense is usually short for valet de chambre; the general sense of "male household servant of the meaner sort" going with the variant form varlet. First recorded use of valet parking is from 1960.
So the idea is one of "Parking by house manservant", with car parking being something that is analogous to one of the regular duties a house manservant would have had. This suggests to me that valet parking evolved as an evolutionary thing. It's easy to imagine a progression like this:
- The valets of a noble house provide services to the house's guests that require knowledge of the house, and therefore couldn't be done by the guests' own travelling staff, such as stabling
- Establishments begin to provide equivalent services that a visitor to a noble house would expect from that house's valets
- This creates an expectation that certain things (such as stabling) will be done as a valet service - something that is done as a matter of course, not as a formal advertised business.
- These services re-emerge as a type of named, formal advertised business when you have logistical factors that require that extra level of named accountability (such as trust with car keys), non-commonplace skills (as driving would have been), and abstraction from the immediate premises, e.g. when a valet service might serve multiple establishments in a city location, when they might need business-to-business relationships of their own with multiple 3rd parties such as if they need a network of possible secure parking locations (sourcing parking spots probably wouldn't have been a simple matter when such services first emerged)
This would explain how it sort of slides into existance around the 1930s in an undocumented, uncelebrated way - as a formalised development on something that was commonplace and informal at the time. This would also explain why not much would have been written about it, if it emerged as a improvised business model trying to implement an adjusted version of something that was familiar, rather than a classically notable innovation creating something novel.
Hopefully that's a useful starting point and a plausible theory suggesting leads and which can be tested and explored.