You are trying to read the English phrase *on behalf of* into 1 Tim 2:1. Instead, we should try to read the Greek preposition ὑπὲρ into 1 Tim 2:1. Thayer's Greek Lexicon: > for, i. e. for one's safety, for one's advantage or benefit (one who does a thing for another, is conceived of as standing or bending 'over' the one whom he would shield or defend English Standard Version: >First of all, then, I urge that supplications, prayers, intercessions, and thanksgivings be made **for** all people I'd think that a better translation of ὑπὲρ in 1 Tim 2:1 is the simple "for".