It's possible, but some important differences between Stealth and Perception make it questionable
The rules on passive checks give us some guidance here. As you mentioned (bold added):
Such a check can represent the average result for a task done repeatedly, such as searching for secret doors over and over again, or can be used when the DM wants to secretly determine whether the characters succeed at something without rolling dice, such as noticing a hidden monster
Although this isn't necessarily an exhaustive list of when Passive Perception would be used, it's the best guidance we have. So let's go through it point by point.
Average Implies Many Tries: What About Failures?
The checks that explicitly have passive versions (Perception, Investigation) are ones which you are essentially taking myriad times every second. You are always noticing hundreds if not thousands of things, most of which your mind edits out. And you are always thinking, even if you try not to. In this sense, they are similar to stealth, since stealth is more of an ongoing process than a single moment's effort.
However, where Perception and Investigation differ from Stealth is that many of these myriad checks can fail without causing the overall effort to fail, as long as some of the checks succeed. If you are walking towards an ambush, and for two seconds don't notice it, you've still "succeeded" on perceiving it if you then notice the ambush on second number 3 (and can react to it before the trap is sprung). However, if you stealthily creep forward for two seconds, and knock over a shelf of fine china in second 3, then stealthily creep forward on second #4, I think it's safe to say that your subsequent stealth won't make the effort a "success."
To summarize, when a passive check is used to represent an average result, it should be done when a failure on the check would not make a subsequent effort harder. If you fail to pick a lock, you may break or twist its mechanism. If you fail to convince someone of something, they may get tired of your attempts to sway them. But if you fail to see something, that doesn't make it less visible later. And if you fail to be silent, you can't make someone un-hear you.
What about a hidden observer?
The second point is a more viable one. A passive check could be used when a DM does not want to reveal to players that there is something happening which could succeed or fail. It's totally valid to suggest that there are times when a character might be observed secretly, and the DM doesn't want to let them know. Perhaps a seemingly inert gargoyle above is really a careful sentry, or an invisible guard is on watch. For whatever reason, it's quite possible that someone could be listening or looking for a character without them knowing.
But most of the time that a character is being stealthy, they only think they might be observed. If you only attempted to be stealthy once you'd actually seen a guard, then the guard would usually have seen or heard you already as well. Rather, most times that people attempt to be stealthy, they know that there might be an enemy ahead, and respond accordingly.
So although a DM certainly has the prerogative to assign a "passive stealth" to a character (so they can hide the fact that an observer is in the area), they risk blurring the line between situations that merit a passive check and ones that do not. And in doing so, they risk setting a "floor" to the standard stealth check, by basically allowing a passive check most times an active one would also be valid. We'll go into this in our next section.
How stealthy must you be to be stealthy by default?
Running with the "unseen observer" angle for a moment, let's assume that a character (who for the sake of simplicity, we'll assume is a Rogue) is not in a situation where they think they might be observed: not in a dim dungeon or mysterious cave, but rather walking down the street of a familiar village, or climbing the stairs of their own home. In these situations, a Rogue would be unlikely to be making active stealth checks. But a passive check could be useful to a DM, to hide the lurking menace and maintain the surprise.
Now, this argument assumes that a stealthy character is stealthy by default, which is not that much of a stretch. After all, it's easy to picture how a trained Rogue might be light on their feet during everyday activities: making only a muffled footsteps as they traversed their own home, or seeming to appear suddenly to a group of friends. These activities might be second nature to a Rogue, the same way that a brawny Fighter might always lift heavy loads with athleticism and grace, even if they weren't trying to show off.
However, it's worth asking exactly how trained or experienced a Rogue needs be to attain this particular level of mastery: where even without trying, they are notably competent at their common skills. And there is an in-game answer to this question: 11th level.
By 11th level, you have refined your chosen skills until they approach perfection. Whenever you make an ability check that lets you add your proficiency bonus, you can treat a d20 roll of 9 or lower as a 10.
An 11th level Rogue gets this powerful advantage. To assign a Passive Stealth result before 11th level, then, might be over-valuing the skill level of the character. At the very least, it would be stepping on the toes of an 11th level ability, many of which provide class-defining advantages (especially since there's no way that one character can have 11+ levels in more than one class). As such, giving someone an ability that is similar to an 11th level class ability (which they have not earned) is something I'd highly advise against, without serious thought.
You might be missing out
If you are the DM of a game, and you think that a passive check is warranted, then you are correct: it is the DM's prerogative to call for checks, active or passive, at times they deem appropriate.
But it's worth asking yourself what the passive stealth check is for. If it's for determining the average result of many stealth checks, then consider that those "many" checks may have some failures in them. And if it's for hiding the need for a stealth check from players, keep in mind that stealth, by its nature, is usually done when its need is uncertain.
And perhaps just as importantly, ask yourself what you gain and lose by a passive stealth check. Consider that a "passive perception" check will prevent the players from feeling like they are missing out: after all, if you call for a perception check, and then declare they see nothing, a player will likely feel cheated (perhaps there is something here, but I'll never know). But if a player is asked to make a stealth check, and then told nothing happens, how are they likely to feel? I suspect that the check will either add to a feeling of competency (if the player believes they succeeded) or a feeling of suspense (if they believe they failed). Either of those feelings is a valuable addition to many games: so ask yourself if it's worth losing out on them before you decide on a passive check.