They shouldn't do anything different than what they're currently doing. Let Uncharted be Uncharted, it's not Tomb Raider. Anything not involving shooting needs to be simple and painless, it must not hold the player up. I know this offends some people's sensibilities, but Naughty Dog know better than they do. This is an shooting game with pretty interludes to set a pulpy action/adventure mood and I'm sure that's the way it will stay.
The "adventure" segments are fun additions to a shooting game. Playing Uncharted with other people, I find these parts are total crowd pleasers, and as a player it's nice to get a break. I love the atmosphere and sights during them, and they don't take up enough time to make it an "interactive movie", the meat of the action gameplay easily holds up to any other TPS. If I want to solve a challenging logic puzzle or some irritating jumping puzzle, I'll play another game designed with that kind of thing in mind.
All I want is for them to add another 15-20 minutes of this adventure-lite content, something like that extended jaunt on the first island in UC1, or the village/ice caves of UC2. UC1 in particular felt like it needed a chapter like this later on, given how much of it there was early on.