RAW, active and passive describe the rolls, not the characters
First off, let's deal with a common misconception. Active Perception is not when a character 'actively looks' for hidden monsters, it is when the player actively makes a Perception roll. Passive Perception is not an intuitive sense the characters have even when they don't search, it is when the DM determines a result by using the passive Perception score of a character without the player making a roll. Thus, 'active' and 'passive' refer to what the player and DM are doing in the meta-game, not what the characters are doing within the narrative, and you should not be determining whether the players 'get' an active roll based on what they say their characters are doing, but rather on how you as a DM want the meta-game to proceed.
The purpose of passive checks is explained in the PHB (emphasis mine):
A passive check is a special kind of ability check that doesn't involve any die rolls. 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
If your players have a dedicated scout, and both you and they are comfortable with that character's player making active Perception checks (that is, rolls) upon entering a new area but not metagaming the results, active checks are perfectly appropriate. That is, a scout doesn't know whether she is not seeing monsters because there are no monsters, or whether there are monsters there but they are very well hidden, or whether there are monsters there and they are hidden poorly but she looked for them even more poorly. The characters know the result of an action, they don't know the result of a player's roll. If your players are capable of seeing a poor roll and having their characters walk into the room trusting that it has been scouted (despite their own meta-knowledge as players), then go ahead and call for the roll. But if your players exhibit the behavior you described - the scout rolled low so suddenly everyone wants to make a check - then as a DM you are well-advised to switch to passive checks. Your players have shown that they can't be trusted with meta-information about how 'good' a roll was, so you simply don't give them that information. By definition, passive checks are used "when the DM wants to secretly determine whether the characters succeed at something without rolling dice."
A second misconception is that passive perception is 'always on', that it is an intuitive sense that represents the absolute worst awareness a character could have in any situation. In fact, the rules assume that a character is always actively looking for danger (the character is active, even when the check is passive) - otherwise they would not have survived this long in a violent and dangerous profession. Even though a character is always actively looking for danger, they do have limits - limited attention and limited time. Passive perception checks should not be allowed when a character is busily engaged in something else. As the PHB explains about Noticing Threats (emphases in the original):
Noticing Threats Use the passive Wisdom (Perception) scores of the characters to determine whether anyone in the group notices a hidden threat. The DM might decide that a threat can be noticed only by characters in a particular rank. For example, as the characters are exploring a maze of tunnels, the DM might decide that only those characters in the back rank have a chance to hear or spot a stealthy creature following the group, while characters in the front and middle ranks cannot...
Other Activities Characters who turn their attention to other tasks as the group travels are not focused on watching for danger. These characters don’t contribute their passive Wisdom (Perception) scores to the group’s chance of noticing hidden threats. However, a character not watching for danger can do one of the following activities instead, or some other activity with the DM’s permission...[Navigate], [Draw a Map], [Track], [Forage].
Thus, when you are using passive Perception, characters will automatically notice threats based on their scores - so long as they are in a physical position that allows them to actually perceive the threat and only so long that they are not engaged in another task that requires all of their attention.
In your case, you have a group of meta-gaming players who open a door and ask if they can see any monsters in the room. RAW, you compare the Stealth scores of Hidden monsters to the passive Perceptions of any of the characters who can physically see into the room in their current position and who aren't otherwise engaged in any attention-demanding tasks, and then you report the results. No dice are rolled.
What style of game do you want?
You have told us that you "want to introduce some ambush style encounters in" your game, but that you are concerned you have "created an environment where [your] players are constantly opening doors and implying they want to make perception checks. It bogs the game down" and ultimately with enough players the ambush will be perceived anyway. Further, you want a procedure that is "efficient and fair, but also presents a level of danger for players who are "too careful"".
You can do all this with passive checks. First, you need to establish that the scout can spot most creatures attempting an ambush. Make sure you have a few encounters that are failed ambushes - challenging in their own right, but which could have turned real bad if the scout hadn't spotted the hidden foes and tipped the players off. Choose creatures whose passive Stealth is lower than the party scout's passive Perception, and in your narrative description emphasize that it is only because their designated scout is especially perceptive that the ambush was averted - they enemy was trying to Hide, the rest of the party would have missed the Hidden foes, but the scout was able to see the enemy and alert the party. After a few of these encounters, the party will learn that they can trust their scout's passive Perception to keep them aware of Hidden threats.
Once they have internalized this message, then you throw them a curve - Hidden monsters whose passive Stealth is above the scout's passive Perception, such that there is no way they could know about the ambush. Make sure the monster(s) are such that without surprise, it would have been an easy fight, but the encounter is challenging because the ambush worked. If it was a challenging encounter to begin with and then you added 'automatic surprise' on top of that, the players would likely feel like you had become an unfair, adversarial DM.
Now the players know something else - their scout is usually up to the task of spotting Hidden foes, but some monsters are just too good at hiding.
How they react to that is up to them, and it could change the play style of the game. It might make them "too careful", but that is fine if they enjoy this style. Many old school D&D games were about resource management, cautious exploration, and opportunistic looting - combat was avoided if at all possible. The 'danger of being too careful' here is twofold - the risk of boredom if your players are more interested in action than exploring, and the opportunity cost of spending in-game time being careful if they do welcome that kind of play. Time is also a resource they can run out of.
If you don't think this style is for your group (and you should have a meta-discussion if in doubt), reassure them that the kinds of monsters that rely on their superior, undetectable ambushes are rare. There is always a chance of them, increasing the tension of exploration, but they are actually rare, such that it is usually more worthwhile to simply trust in the passive Perception of the scout. Those are the kinds of interesting decisions the players should be making, risk vs. reward comparisons that rely on them investing in an understanding of how your world works.