Yes, a true Indiana Jones game should combine these elements. But like with everything it's mostly trial and error; you might have a good idea but then who's to say it's still good after it's been developed? Or if others will receive it well.
Since I prefer the classic adventure games over the "Tomb Raider" varieties, I'm leaning towards some kind of 3D adventure game environment, kind of what they did with "Gabriel Knight 3", but with much bigger maps, something similar to a GTA kind of freedom of exploration. Not one huge map, but big enough and different ones for each location/level/chapter in the storyline.
The tricky part really is how to integrate the action/combat elements. I haven't played that many different types of games, so I can only suggest what I know. I've always enjoyed the "Silent Hill" exploration gameplay, though the combat of those games isn't all that great (I'm talking of the games up to "Origins", since "Homecoming" is PS3 and I don't have that). I kinda like the combat and hot key system of "Staff of Kings", it's just a bit chaotic in that game but with the right people on the job I think it can be perfected.
I'm also thinking of some kind of co-op mode INSIDE the main storyline. They did that for the "Obscure" games: if you played alone, the AI would control the second character, but with a simple push of a button, another player can take over the task. Could be interesting if those second characters have their own specialties. Kind of like with the Lego game, but more seriously implemented.
I don't know, I'm just brainstorming. In the end it would just be a cool extra, at least cool if it was worked out decently, but not strictly necessary. After all, I always play games by myself and I think many people do that as well. But I'm not one to call majorities or anything.
And a bit off-topic, but does anyone else hate that many game catalogues and databases group their titles together under action/adventure? Like on eBay, when I want to look for adventure games specifically, I always have to wade through countless of other type of games. Sure, they might have adventure "elements", but they aren't "adventure games".