Another idea that could add some immersive quality could be to try to have visual information from the cards play a game-relevant role. In other words, the illustrations on the cards aren't just artwork to look at -- there could be in-game effects or systems that cause you to scrutinize the cards and activate certain game systems depending on what you find. I don't know if I've ever seen another game that does this, and it makes sense why. In most games, especially Euros, cards have icons, numbers, text, etc, to specify exactly what a card's function is, and it's always the same, every time. But the idea of cards having variable effects, depending on the players' attentiveness, seems to fit well with the theme of this game.
Nevertheless, this idea isn't completely new. In a pretty early version of the game, the lost temple was actually a series of cards, each of which showed a room of the temple from the adventurer's perspective, and each of which had several "features" that the players could test, like ropes, levers, and exits; and a solution table would tell you the result of testing each feature. Several images and icons were present on each of the cards, and the clues would correlate to these. For example, a clue might tell you to look for "baboon" symbols to help you find your way to the grail, and so if you saw a baboon symbol on the wall of a room you were in, you might conclude you needed to test the feature on that wall to see what it does. This was all kind of cool, but horribly cumbersome in practice. And we've since abandoned a fully-constructed temple anyway, in favor of a randomly assembled temple, for ease of setup and playability.
But the idea could still fit. My most recent thought sprung from looking for ways to boost the sense of the enemy pursuing you, beyond the timing track. One neat effect would be if you could realize that a certain enemy operative was following you. And I realized that a way to enable this could be to incorporate this effect into the illustrations on the encounter cards; some of them could include, lurking in the background somewhere, various shady characters; perhaps if you happen to notice the same shady character on multiple encounter cards, you can announce that you've discovered that an enemy operative is following you, and then weaken the enemy in some way. Or maybe it's that there are several "enemy operative" cards, showing their image, and when you see that operative on an encounter card, you get to take some special action or whatever.
Separately, I thought a fun way to integrate ideas like Indy's fear of snakes is to give each player-character a "phobia", and if another player spots your phobia on an encounter card that you face, perhaps you automatically fail the challenge. Maybe this could add some fun, as well as some incentive for you to pay attention during other players' turns (and the clue system amplifies this).
This is different than the way most games work, and I suspect it hasn't been tried because it opens the door to the rules lawyers to come in and suffocate the fun of the idea. Specific considerations like "how long are you allowed to look at the card to search for an enemy operative?", "does the active player have to pass the card around for all of the other players to scrutinize?", etc will invariably pop up. And that also doesn't take into account the consideration of memorization; once you've played the game enough times, you may start to memorize the various encounter cards and know which operative is shown on which, or which phobia is present on which.
[Incidentally, coming back to the "enemy operative" idea -- this could instead be a different way to actuate the "ambiguous character" lever. In Last Crusade, you see these guys with fezes pursuing Indy and Elsa but you don't really know who they are; come to find out they're actually good guys (more or less). We could achieve this by dealing out, say, 3 "ambiguous character" cards face-up, each with a picture of that character, and dealing each a "true identity" card face-down next to each. During the course of facing encounters, if you spot one of the ambiguous characters on an encounter card, you can undergo a mini-challenge to confront that character and reveal its true identity, which then has some effect on the game. Of course, this doesn't really work for major-character ambiguous characters like Elsa, the Maharajah, or Mac, but only for minor-character "people who are pursuing Indy" ambiguous characters. Presumably having two tiers of ambiguous characters in the game is putting too much game-mechanical weight behind a system that isn't the main focus of the game. But again, it's simply something that this idea could accommodate seamlessly.]
So there are practical issues with implementing an idea like this, and making it even more challenging, it so strongly depends on the quality of the artwork that it will be nearly impossible to playtest without first commissioning the game art. But I like the potential of the idea, and it could find its way into other games as well.