There are two sides to the issue: playability and verisimilitude.
In my experience, if you have a thief in the game they'll search every door for traps. It slows the game down. You might forestall this my not putting traps in the game, but then you nerf the thief by not letting him use a core ability. So you put traps in so the thief will have something to find, but now he needs to search everything. It's like a solution looking for a problem.
Back in the day, people handled this by establishing a "standard operating procedure" for doors and chests, along the lines of: check it for traps, poke it with a pole, listen for noise, all while standing back as far as possible. Since it was onerous to do this for every single door and chest, the thief would inform the DM upon entering the dungeon that he was "searching constantly" and that SOP would be followed at every door. So the end result was that the DM applied the thief abilities passively. I played in multiple groups that did this, it was the rule rather than the exception.
(The most extreme example was a group that used a cable and winch to stand back 50 ft and pull every door out of its frame...)
In terms of versimilitude, the thief has a worse chance to find traps than a non-thief until third level! That means having a low level thief in the party would actually reduce your chances to find traps. Also, it only stands to reason that anyone could attempt the same things the thief does. What makes the thief special is that he can do it better, faster, or or with fewer restrictions.
- Anyone can open a lock, the thief can do it without a key.
- Anyone can pick a pocket, the thief can do it without getting caught.
- Anyone can move quietly, the thief can move silently like the undead.
- Anyone can climb a tree, the thief can climb a sheer surface like a glass wall.
Anyone can climb a wall with a rope, the thief can free-climb.
- Anyone can hide behind a barrel, the thief can hide in plain sight if there are shadows.
- Anyone can attack you from behind, the thief is more accurate and more deadly.
- Anyone can find a non-magical trap (1/6) if they explicitly search, the thief can find magical traps even when not searching.
Some of these abilities verge on the mystical and that is a big part of what makes a thief special. But note that the thief's ability does not exclude doing it the mundane way. The thief can still use a key on a lock, use a ladder to scale a wall, or hide behind a barrel. What if he hides in shadows, behind a barrel? What if he tries to open it with key and fails (because it's not the right key) and then uses his open locks ability? He just made two attempts, one mundane and one skilled.
For these reasons, and because of the probability issue, I give the thief two attempts to find a trap: one by searching for one turn in the correct location, and one by using his special skill.
Regarding traps, there are several conditions mentioned for all characters: you have to take a full turn, you have to say you're searching, you have to be looking in the right place, the trap has to be non magical, and you only get one chance to find it. For the thief, it only says that you only get one chance to find it. The fact that one of the restrictions is repeated suggests that the other restrictions are not applicable.
That implies that the thief can find any trap, magical or otherwise, even when he does not say he is actively searching, and he finds it instantly. That essentially defines "passive detection".
So when DMing that is how I do it. I roll for the thief to detect traps passively, sort of a sixth sense for such things. If the party still wants to search for traps, the thief is welcome to join in and then gets another 1/6 chance to find it like everyone else. In this way the thief is always a benefit to overall searching ability, and has the additional benefit that he may detect something nobody expected. It speeds up play, and it makes the thief special and valuable.