So for this example, I'd approach it from the opposite direction.
Your rogue is trying to get to the Altar of Doom, and a couple turns in the head priest realizes that there are only three people fighting, when there are supposed to be four.
"Where's the fourth interloper? Everyone! Some little rat is trying to stop the ritual! Find him!"
And now the enemy are making Search checks to try to find you. They are trying to beat your Stealth check, though you might get a bonus if you manage to stay behind cover.
That, to me, feels better than a constant series of Hide checks. The threat is the enemies actions, and how much they can dedicate to Searching vs remaining engaged with the fight. If they Search, it's harder for you, but easier for the rest of the party. Likewise, the rest of your party can do things to keep the cultists' attention on them, and unable to dedicate time to Searching.
Essentially, it's a dynamic that involves everyone, not just one person rolling over and over. The Hide roll is just the start of the action, not the action in and of itself.
And this is indeed a good question.
I would say that if you are clearly visible, then you are found if you are identified as an enemy or target (as I described in a previous post). If you are not clearly visible, then it takes an active Search check. If you are not identifiable, then it takes an active Search check.
That said, there are lots of things I can see affecting that. For example, what about someone affected by an ability that causes them to have disadvantage if they attack anyone by the source of that effect? (For example, the barbarian's Lion totem.) In other words, it forces a near-exclusive focus on the source of that effect. Would that not also apply disadvantage to their general awareness?
What about people who are getting knocked around in combat? The Push mastery, or the barbarian's Forceful Blow from Brutal Strike, or a warlock's Repelling Blast? Or even just being grappled? Those all seem like things that would make it much more difficult to see what's going on around you clearly enough to identify a skulking individual.
Basically, there are lots of things that can impact an enemy's ability to Search or find, and that increases the involvement of everyone at the table. There are not so many things that impact the basic Hide, and that check is largely isolated to the person hiding.
So I feel like it's a worse choice all around in terms of rules.
We've done a lot of white room discussion in the thread, but not a lot of thinking about how the players' actions interact with the rest of the group, and what leads to more interesting play.
Well, sort of, but not entirely? You're using the Hide action to gain the Invisible condition, not merely to "hide". The direct benefits of that condition are that you gain advantage on attack, and enemies have disadvantage on attacks against you. The implied benefit is that the enemy is not aware of where you are, or hasn't identified you as a threat, and thus isn't able to defend itself as well, or react in time, or bring force to bear in a proper manner.
Thus the Hide action is to put you in a state where the enemy is not sure of where you are, or doesn't recognize you as a threat, allowing you to get the jump on him in the event of combat. This is primarily done by not being seen or heard, since doing things that draw attention to you (making noise, casting spells) end the condition. Being out in the open also tends to end the condition, since that makes it easy to "find" you in most circumstances.
However nothing implies that movement will end the condition. As long as you remain out of sight and don't make noise, you can remain hidden from the warehouse guard all day, following him around the shelves.
On the one with the guards coming down the hallway, that would work mostly the same in either interpretation. You duck into a side room to Hide, hoping they don't notice you. After that you can let them pass and continue on your way, or attack them. But you could also slip out of the room after they pass and follow them. The continued movement wouldn't end the state of them not being aware of you.