Well "thee" certainly makes one think of Old English, conjuring up images of Shakespeare and the King James Bible. And very interesting when combined with John's observations about the history of latin gangs. Graffiti art is one of my personal interests (not as an artist, but as a fan of the medium) and the old english script has been a staple of latino gangs since the 1930s or 40s with the zootsuiters . Early so-called "Cholo Graffiti" pieces are called 'placas' and are characterized by old english letters, almost always in all caps, and traditionally in a formal business-esque structure of headline, body & logo, such as: gang/street name, roll call & signature or tag or the artist. The formal structure and typeface was meant to convey a seriousness about the gang to illicit respect and exclusivity.
A little bit of internet searching shows the word "thee" at use in early chicano gang names, too. Luis Rodriguez talks in his book "Always Running" about setting up a clica (gang or club) at 11 years old, in the early 50s in Las Lomas, CA called Thee Impersonations. He says: "the 'Thee' being an old english usage that other clubs would adopt because it made everything sound classier, nobler, badder." (his emphasis, not mine)
So maybe these early LA garage bands were just emulating what they saw in their environment... it may have afforded both protection and patronage, as well as provided a clue to their local pride or credentials - giving them safe passage, if not acceptance, because they're also a local. Remember too, that latin rhythms are often cited as a component of early pre-beat garage styles. hmmmm....
An interesting story is starting to pencil out here, but I feel like there are more details to fill in. This latino gang connection definitely feels solid. So far, it looks like chicano clicas must've first popularized the use of old english references, to infer formality and induce respect. Then local bands either copied or adopted the trend for a variety of reasons, but mostly self-preservation. The usage must've had a bit of "coolness" to it as well, since it continued to grow from there. I'd like to find a copy of that Lux Interior source mentioned above; it might supply more clues.
A slight tangent here, but this is all pulling me towards another thought, too: I wonder if similar connections or inferences to gang culture could be the source of the "bad ass" persona in musicians. Up until the 50s, most musicians had a stage presence that was either a very jovial, party-time, funster type of schtick or a very serious, traditionally studied and formal demeanor. Even early pre-rock & roll styles employed the fun & freewheelin' persona. But then something changed in the 50s, and a stylized "outsider" type of character develops: disaffected, rebellious, and decidedly non-conformist. Not only in music, but film and books, too. I wonder if fascination with early gang cultures could've given us this collective attitude adjustment as well.