I agree with the comment that it's unclear what the prescription to ignore the fact that bridge is played in partnership means. You've correctly calculated the number of ways a bridge game can start if you regard the players as distinct, which is the appropriate way to look at the situation in bridge.
To illustrate how the result can depend on what you regard as different situations, consider the game of Getaway, where the player who holds the ace of spades starts the game. If you want to know how many different situations you might face as a player, the result would again be the same (assuming four players are playing), since each player is in a distinct position relative to you. However, if you're not one of the players and you don't care about their identities and just want to know how many inequivalent situations might arise in this game, you need to take into account the cyclical symmetry among the players. The order of the players still matters, so there's no full permutational symmetry among them, but it doesn't matter whether you number them e.g. $1234$ or $2341$. Thus you'd have to divide your result by $4$ to get the number of inequivalent starting positions. This doesn't apply to bridge, though, since in bridge the dealer opens the auction, so there are distinguished first, second, third and fourth players. Thus, in the case of bridge, your result is right no matter whether you look from the position of a distinguished player or from outside without regard for the identity of the players. Since this is true with or without partnerships, it's not clear what the prescription to ignore the partnerships means.