It wasn't theologically neutral, though. To address God with the polite plural form of the second-person pronoun suggests that the speaker is addressing an entity that occupies a higher position in some kind of social hierarchy, which implies that one's relationship to God is mediated by social structures. Use of the singular form was considered more consistent with Protestant theology, which emphasized a direct and individual connection between the believer and their God.
You see the same phenomenon in German, a language that's retained multiple forms of the second person. In German translations of the Bible, and in German prayers, God is addressed as "Du" (the familiar singular "you").
You also see it in the Latin Vulgate, where God is addressed as "tu" (singular), rather than "vos" (plural). My point is that the translation as "thou" was more-or-less a direct translation of the original text in almost all cases, preserving the distictions made in the Hebrew and Greek manuscripts. At the time those English translations were made, the choice was theologically neutral insofar as the original texts were theologically neutral.
At the time James VI/I commissioned his Authorized translation, "thou" was already starting to fall out of production, but was not something that seemed foreign or archaic to English speakers, and he'd asked his translators to follow the style of the existing English translations, which were by-and-large based on Tyndale's. There are certainly some ways in which the translation attempts to emphasize certain theological/christological viewpoints, but the usage of "thou" is not one of them.