First of all, a skill check isn't automatically a success just because someone if proficient in it. It just gives you a better shot. So failures will still happen a *lot*, rest assured.
As for different background dialogue options: A cleric in the party absolutely should be able to chime in when you're conversing with an NPC about cleric stuff.
DnD is a party-based game, where most people would aim for a balanced party of different specializations, and you usually have group conversations by default. It's very unnatural to exclude the rest of the party from the whole conversation.
Finally, you usually don't cover *all* skills even in a 4-player-party, so there's lots of room for replayability or different approaches still.
