Does this come up often for you, that a PC is sneaking around, and a hidden NPC is searching for them? I can’t say it does for me. I would imagine the NPC would be too focused on remaining hidden to also be searching. If players have to pick one task or the other when exploring, shouldn’t NPCs have the same limitation? I don’t know, doesn’t seem like a major concern with the way I set up challenges.
So I'm a big believer that you can do one thing well. If you are focused on sneaking about you can not also be focused on searching out for hidden enemies. The most likely outcome when you have two groups focused on not being noticed is they pass each other like ships in the night. They might notice each other passively, but generally only if someone rolls poorly.
Isn't the point of scouting to gather information while remaining unnoticed? I find the idea that one can't perceive threats while being stealthy to be rather bizarre. Even the 5e travel rules don't prevent a party moving stealthily from using their passive perception: you only get denied that if you're mapmaking or foraging or the like.
And passive perception is defined in the rules as the DC for stealth checks. The odds that an opponent successfully evades your notice is not affected by whether or not the oberserver is taking the search action. The action just gives the observer another chance to notice the sneaking character.
I would also note that insufficient lighting can obscure a potential observer from the character being stealthy. A potential observer out of hearing range, standing in a darkened room and looking out a second-story window, is not going to be noticed by a character trying to use stealth to be unnoticed as they move through a crowd below. Similarly, the stealthy character may be within visual range of the observer, but not vice-versa (e.g. observer is an Eagle-Totem Barbarian).
And yes, while the original question wasn't aimed at me, in my games this comes up all the time. Locating the enemy before they locate you is critically important to being able to engineer an encounter on favorable terms (or avoid an encounter altogether).
As a typical example, consider a scenario where the party is trying to approach an enemy fortress unseen by either the scouting patrols or the occupants of the fortress. Failure would be devastating: the enemy will know they are under attack, and can quietly prepare their defenses, making what might have been a series of manageable, separate encounters into an unwinnable mass brawl. But unless the party loactes all the scouts and has a way to observe the occupants of the castle, the party will never know if they've been successful or not until the trap is sprung (or they reach their target and find it unawares).