In 2e, traps are there specifically to slow down the party and force them to be cautious. Even in 5e, traps are a break in the action and can be triggered before anyone sees them. The 2e style doesn't sound like your playstyle at all, but even the way 5e uses traps by default seems to not be your style either. So...
Tell them the trap is there
Traps are interesting when players interact with them. Tell them the trap is there! Yes, 5e has Passive Perception for this purpose, but don't let that ruin your fun if giving the check away "for free" will significantly improve your and your players' enjoyment of the game. (Passive Perception is still useful for surprise encounters and ambushes, after all.)
By telling them that a trap is present without telling them exactly where and what, you get straight to the interesting interaction, as they investigate the trap to figure out where it is, what it does, and how to circumvent or disable it. They may play it smart or they may make mistakes and suffer the trap's consequences. Some traps will be boring (and quickly dispatched) and some will be super-interesting (and be super-interesting). Either way—smart or disastrous, trivial or super-interesting—you are keeping the part of traps you find fun (the players messing around with them), eliminating the part you find frustrating (caution bringing the dungeoncrawl to a literal crawl), and quickly moving along to wherever the fun is actually to be found.
So skip straight to the fun, and tell them that there is a trap here. Let them figure out what kind and where exactly, but tell them:
As you move to cross the threshold, your keen adventurer's survival instinct tells you something is wrong. There's something not quite right about either the door's stone frame, or the room's floor… or maybe it's something else nearby that's off. Anyway, what do you do?
...and cue the flurry of investigation!
DCs for the traps, should then be converted like most DCs in 5e. 10 for easy, 15 for moderate, 20 for hard, 25 for very hard.
Best Answer
There are a few different options, and the links in the comment contains a lot of good content.
In this case there are a few things you can do. The player who is scouting ahead, you can either ask them for a perception roll, and say that when they're scouting you'll let them know when they need to roll, or you can let the trap happen and get the player because their passive perception isn't high enough.
I would personally ask for the roll, they said they were scouting ahead, and even though they didn't say they were looking for traps, it's part of what they likely meant.
The reason that I'd ask is this allows you to be the one who calls for trap rolls. You want to do that so that they don't end up with trap phobia. If they randomly stumble across this trap and even if it just gets the scout who is up front, they are going to be worried about traps all of the time. Even if they fail the roll and the trap ends up getting the scout, you've now set that you'll call for these rolls.
That does mean, when there are traps in the future, you'll need to call for rolls from whomever is scouting. With a high roll, they'll find a trap, and if they don't, the trap gets sprung.
You then limit who and when they can roll for traps which are often pretty problematic, again see the comment by @Rubiksmoose about curing trap phobia.