Public prayers are for the public, so it is reasonable to use what the listening public considers to be "formal" language if that's what the group is expecting. Otherwise, they are thinking about the pronouns used in the prayer, rather than its content - and that never should be the focus of attention during a prayer.
However, behind closed doors, so long as the
prayer is honest and the attempt to connect to God is sincere, I don't believe the pronouns matter much.
Frankly, I use both the "you" and the "thee" pronouns fluently, so I say whatever comes out when I pray privately. Sometimes, I don't even bother addressing Heavenly Father, since I believe he knows I'm praying to him.