If the guy doesn’t have the guts to propose maybe he can rely on the shadchan but normally in most circles…the couple knows already before he proposes that they’re getting engaged and the guy proposes to make it official..
The actual source of proposing I don’t know..but if a guy proposes in a meaningful way it’s not goyish…
The proposing with bending down and asking will you marry me? is goyish and should prob. be avoided