I did until I saw what you wrote in your post after that.
I disagree with your second paragraph. In my experience it's more up to how much I enjoy the way the game fundamentally plays combined with the atmosphere/feel of the world more than anything else that dictates how much I explore.
Witcher 2 didn't inspire me to explore, I enjoyed its atmosphere but not so much the gameplay.
Fallout 3 and Vegas inspire me to explore a heap. Those games combine great gameplay and great atmosphere imo which is why I'll go and check out pretty much every nook and cranny. Even something like a lone bit of unique writing on a wall or a great vista is enough reward for me in these games.
HR's gameplay grabbed me just enough to get me to critical path the game in the most lazy way possible which is in contrast to its atmosphere which I loved.
I spent nearly as much time just finding shit to blow up in Just Cause 2 as I would completing a massive RPG purely because I loved every minute of the gameplay, traversal and action both done so well imo.
What do you think needs to be done to make exploration interesting in these types of games if it isn't the base gameplay and/or atmosphere?
Since we're just talking these types of games, I'll use this Deus Ex and Thief as the primary examples, I think the most important thing is the stealth has to be top notch. I don't feel like it's as good as it could be in HR and Dishonored and I think the main reason is because there are too many powers and over powered abilities that the gameplay is designed around and it makes the game easy.
The balance being ruined also affected HR's exploration because they reward you
too much and 70% through the game you can have just about every ability and have some of them leveled up a bit too. There's nothing really useful to buy except praxis kits because everything else can be scavenged easily, mostly because the level design is not as crafty as it was in Deus Ex (the fully detailed map hurts that too). So if there's nothing to buy and you're rewarded too much, optional exploration becomes pointless because you already have everything you need. As far as exploration in terms of completing objectives, I've said many times they didn't do a good job there in the level design. Most rooms can be completed with a vent or a hackable door.
The way I would fix these modern problems in Thief 4 is simply don't have any kind of fancy powers. Give me a bow and a sword and other stealthy items (example being the water arrows to put lights out) and actually design the levels in a clever way. I know that requires a lot of effort and skill on the developer's part but we're in the big leagues, no whining allowed. I also know that without fancy powers you can't market it to the casuals and try to trick them into thinking it's a different game than what it's trying to be. Also without powers there won't be a risk of not balancing the level up system and making your guy Superman. As far as too much loot goes that can be solved by simple numbers. Like instead of putting 10 health potions in a level put 3, or instead of giving us 30 water arrows in a level give us 10. Something along those lines.
It's not easy to do all that, I know. But Deus Ex and Thief 2 weren't created because they were easy to make.