How can I make skill checks more than just rolling a die?

As long as its card based, you can tweak the probabilities easily.

The perception check you proposed works really well with cards. Depending on the perception level, there can be lesser blanks in the search pile. Of course the down side is, there is a limit.

Not sure how the intelligence would work with an IQ test, and I don't think players would be willing to do a cartwheel or a split for the dexterity check.

But essentially, probability through cards may not be any less random than dice, but can FEEL less random than dice.

Its down to the experience you wish for the player to experience through the new interface.

Mansions of Madness has a fun little mini game which acts as an "intelligence" check of sorts. They have puzzles which are either A: unscramble these tiles and build the picture shown; or B: close the circuit (sort of like a pipe game).

All players can attempt these tests, however the amount of moves you are able to make to solve the puzzle depends on your characters intelligence stat. So fighters with 2 intelligence get 2 moves to make on the puzzle, often needing several turns to finish. Intelligence characters with 8 intelligence can usually finish the puzzle in one turn (as long as they know how to solve the puzzle).

There is an example of one of the many puzzles in this video at the 5 minute mark: https://www.youtube.com/watch?v=h8OQyVjU07A

And this is an example of one of the easier arrange the image puzzles, which you can probably use for perception checks. The puzzle starts at 2:40 mark: https://www.youtube.com/watch?v=ZJ3tR9JZGQ4

For perception, lets say the situation is finding a switch, or trap, or loot, etc, we'll call this the objective

So there is a card representing objective, maybe there's a tick on the card

There's a second type of card which is basically a blank, maybe there's a cross on it.

So when a perception check is called, a pile made up of 1 objective card and some blank cards is made.

The perception level decides the number of blank cards in the pile. Maybe by default there are 10 blank cards. If your perception level is 4, 6 blank cards are put in the pile instead.

The player will shuffle and choose 1 card from the pile randomly. If the objective card was drawn, the perception check is a success. If the blank was drawn, then it fails

This can be tweaked in so many ways to fit the needs of the game, and doesn't even need to be restricted to perception.

...

And Despot9 pretty much sums up the next issue, does how you fulfill a check even matter, since role-play is more about story and interaction.

I'm imagining a LARPer wanting Gladiatorial Combat to the death because Rock-Paper-Scissors was too boring, as an exaggerated example of this topic

To be honest, I agree I'm not fond of the idea of minigames for the exact same reasons. You limit the number of replays and risk balance by making it possible for the players to "Know" the answer.

I'd probably simply give the player more influence on the results. At the most basic, allow the player to "Exert Himself" before a roll to improve his chances. Every 1 Health he willingly spends, he gains a +1 value.

Or maybe have another player read the "Event Card" with the specific requirements of success. The acting player may spend cards or items while HOPING that he completes the requirements on the card.
While this would have the same memorization issues you could theoretically have multiple requirements for the same event.
For example, "Stranger Approaches" could have one card that requires Charisma to be given a treasure. But another "Stranger Approaches" that will damage a player if he isn't readied for combat.
A player COULD be prepare for both results, but he'd be spending extra resources. That may offset the issues.

I don't like the idea of mini games because it slow down the game way too much.

I also find that in adventure games, sometimes you only draw a card, something happens to you, make a check to survive and that's it. I think the problem is the lack of decision involved in the whole process.

There could be various ways to solve this:

A- Give a choice. Make each event or exploration give a choice between 2 different check for 2 different outcome. You could even hide the check from the players so that he cannot predict which one he has better odds.

B- Use cards instead of dice. Use cards with stats and value. Player will be allowed to keep in their hand stronger cards for later, but if they have no card, they fail the check. So they must now decide when or not to fail a check by keeping or spending the right card. You could also allow cards to stack (spend 2 cards of the same stats to combine value together). So that weaker characters could still use certain cards.

C- My dungeon quest variant example: In one of my DQ variant, if you fail a roll you get a determination token. But you can spend X token to increase your die value by X to succeed the roll. So you give the player a choice. If I really need to pass this test, then I'll flush some tokens, but if failure is not much a problem, I could fail the test and gain a token that I could use on a later turn. So in that case, you make the player evaluate the importance of the failling/succedding the test.

Of course, you could combine many ideas above, and there could be many other variant possible to the suggestion above.

I think I should start using some of the ideas in my own games.

Maybe for perception you could do that tile test you also get on IQ tests, with a gap in a shape, with four possible shapes to fill the gap(rotated to different degrees obviously, to not make it too easy).