okay, I think I better understand where you are coming from. The key to your understanding, from what I read, is the phrase "always on". The idea is that you are always observing your surroundings, and if that uses 10 + bonus, then theoretically if you roll for perception you are technically using 2 results: your always on result and your active result. If both rolls exist, whichever one succeeds means you succeeded. As such, if you roll a 9 on perception but have a passive of 13 and the DC was 12, it doesn't matter that you rolled a 9 because your passive succeeded so you succeeded since you always get it.
I will explain myself, but for the sake of play, my final ruling will be: no, I don't rule it this way.
I will say, however, that for stealth I do mostly agree. If you stealth in combat, your result is compared to the passive perception of all enemies (you have to have full or 3/4 cover or be heavily obscured or invisible to begin with), and you hide from those you succeed against and fail against those you roll under. Unless the circumstances are special, though, those that spot you will probably yell out where you are to others, so essentially in most occurrences you are rolling against the highest passive perception of the enemies. If you do hide from everyone, though, finding you requires either an action to seek (perception check) or for the enemy to stand it a place where the cover you use to hide no longer hides you (like if you hid behind a pillar but they walk around the pillar). Hiding in combat is complicated, but when in doubt try to get 3/4 cover or great, be heavily obscured/invisible, and/or ask an ally to create a distraction.
If you would like to hear my why, you can keep reading. Otherwise, we will go with that and continue the game Monday morning (it is sunday night for me, in case anyone is in another time zone).
[ +- ] My reasoning
I do not want to rule that passive perception is "always on" for 3 reasons:
First, I don't think it is RAW (rules as written) as nothing in the book says this. It very well might be RAI (rules as intended), but unless the book says something akin to "if you roll less than your passive, use that instead" then the rules don't actually say to do that.
Second, I think this comes from Jeremy crawford for the most part, but even he says "Passive checks are a tool for the DM. The DM decides whether the rule is used at all", meaning players were never meant to invoke the passive checks rule.
Third, I think is isn't the healthiest for the game to always use such a rule. I can see it making traps and secret doors use this rule if players would otherwise develop bad habits and become paranoid. I think It makes the most sense for stealth and perception to be active stealth versus passive perception to become or stay hidden and active perception versus passive or recent stealth to locate. However, for lots of other things, it creates the very feeling that was expressed. "I should have not even rolled." If the rule makes a player regret engaging with the game, it probably isn't good for the game. Bad rolls happen. I hate when I roll bad too. However, if the immediate feeling is regret in saying something, that seems wrong to me.