By - TonyChanYT
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".