I've played both ways...let's call them @
iserith's way and @
Hussar's way...and still do. In fact, I would say I end up at tables who play Hussar's way more often, with people who blurt out "Can I make a (insert skill) check?"
Here's how I see the two approaches contrasting:
Hussar's way has the advantage that it's straightforward and quick. The players don't have to try a bunch of different things. "I check for traps" gets resolved quickly. On the other hand, the 'challenge' for the player (if you can call it that) is simply to remember to invoke items on the character sheet. In the case of traps, the interaction with the fiction is the same every time: "I check for traps." If there's variation, it's solely in the DM's narration: "Yeah, you look up at the ceiling and you see...." (This might be followed by a similar thing for disarming it: you roll Disarm, or whatever, and succeed, and the DM narrates: "Using a 10 foot pole, you snag the trip wire and...) Note that the DM is, in a small way, taking control of the character: "You look up..." "You use a 10 foot pole...".
Iserith's way, on the other hand, can suck down a lot of game time as players try 8 things, instead of invoking 1 skill. This can take even longer
when there's no trap. On the other hand, it asks players to engage with the fiction by invoking actions in the game world, not items on a character sheet. In general, I think a sign of a game going well is when the players are doing things that don't require any knowledge of the rules. "I look up at the ceiling" doesn't require knowing anything about the rules. And it doesn't require the DM to take over the character, unless you count "You see a tripwire" as loss of agency. (In which case I don't really want to have debates with you.)
I (perhaps obviously) prefer Iserith's approach, even when I want to shout at my players "THERE'S NO TRAP CAN WE KEEP MOVING!?!?!?!" But I don't think it's intrinsically superior; it's just a gaming style I find more enjoyable.
Also, you can combine the two approaches:
"Characters who search the area will find the trap with a successful DC 14 Intelligence (Investigation) check. Players who look specifically for traps get advantage on the roll. Players who state they study the ceiling find it automatically."
EDIT: I'll add one thing...
Using either approach, I think it's important to minimize making the 'correct' actions/choices be gates to progress. Looking up at the ceiling (or looking for traps) should make it easier to achieve success, but only in a minor way. If there was some foreshadowing/clue that looking at the ceiling is a good idea, the benefit to doing so can be commensurately larger, but failure to do so still shouldn't be show-stopping.