Short Version:
Maybe P is overwhelmed by bookkeeping and it's distracting him from situational awareness. Help him make a mechanically very simple character without fiddly bits or conditionals to keep track of, so he can focus on making good choices rather than having good bookkeeping. Invite the other players to support P with advice and by being good role models for the behaviour he's trying to cultivate.
Long form answer, with rambling and details.
Back in my very first RPG ever--and also my first time as a GM--I had a player whose poor choices got him repeatedly killed. Let's call him Q.
Q knew the rules and mechanics quite well, but had a very hard time applying them intelligently to whatever situation he found himself in (like forgetting to heal himself as a cleric). Even more than that, though, was his role-playing: he really really liked to role-play his characters, but that got him in trouble because when Q got deep into his character's internal motives the PC would lose common sense and perspective about the surrounding context of his actions.
It got bad. Really bad. Q's second character was killed by the party for betraying them (he had a conversation about his friends over tea with a "nice" lady). At that point I shared Making the Tough Decisions with the group. He studied it carefully, had intense discussions with me about it... and as a direct result his fourth character perished of untempered curiosity: the characterisation "very curious" overcame the common sense "half these items are cursed and my friends are begging me to stop," until the pile of treasure he was investigating yielded up a lethal curse.
After that session I took Q aside and we talked. He knew he had a problem, and he was trying to "get better," but he needed help. I'd noticed that all his PCs so far were mechanically complicated and required in-game bookkeeping: advanced casters and races with lots of conditional features and spell-like abilities to keep track of. So we hatched the simplest possible character build: nothing to keep track of. No "if you're flanking, X also happens," no spells, no per-day abilities. If his character sheet said he could do a thing, he could always do it.
We wound up with a kind of Indiana Jones flavoured skillmonkey (a rogue chassis with homebrew mods to replace things like sneak attack because tracking whether you can deal that extra damage was beyond what we wanted for the build). He wasn't optimised in the traditional sense--but since another PC in the party had straight levels in the NPC Expert class, that wasn't an issue in keeping him relevant in the group. Instead he was optimised for what Q needed: a simple no-bookkeeping character to let him focus on situational awareness and making good choices.
At the end of each session he'd hang back --along with any other players who wanted to-- and we'd reflect on the game: what worked, what didn't. We'd consult (and if necessary research) and come up with what to make sure we did again, and what we'd change next time. (I've since found that any game I run which has some form of this "reflect and plan" dynamic after every session is improved by it.)
In tandem with another player rising to the challenge and being a kind of "teach by example" role model, it worked. A year later Q was successfully running complicated wizard builds with great party dynamics and great depth of character. He was a real joy to work with, and all he needed was to wade in at the shallow end of the bookkeeping pool instead of jumping into the deepest part head-first.
nota bene: My players have tended to treat the group dynamic as one of table-level cooperation between friends. However much their characters may be rivals, at the table they collaborate to tell the best stories, and I'm also one of the collaborators. In groups where players and/or the GM act as rivals at the table level of things, I'm not sure how much my experience will be useful. It sounds like your whole group isn't really on the same page in terms of their desired gameplay experience, and communication isn't really strong. Working on improving the "friends at the table" level of things might help your game in a number of ways.
When things like this happen, I always give my players this chance to clarify/confirm, just like you've shown in the examples. My reasoning for this is simple: the game world and what is happening there is closer and more important for the characters than it is for the players. No matter how immersive your storytelling skills and how much everyone around the table has their mind sunk into the game, it's still just a game.
The characters have 100% of their mental capacity to devote to the action they're in—more often than not, their life (or at least career) depends on it one way or another. The players simply cannot pay the same amount of attention, the pesky real world will always commandeer a significant portion of it.
Another important point is that it is much easier to remember something you've actually experienced (seen, handled, been through) than something you have just heard told to you. Again, this gives characters a better chance of being in touch with the situtation than their players.
More than once, I've said to a player: "Yup, you've forgotten this or that, but I know your character wouldn't have. So I'll tell you." This can apply to a lot of things, ranging from the examples in your question to things like "the name of the tavern where we were beaten up last time we visited this city." Even if they don't remember the name itself, they would most likely recognise the place before they blunder in.
Best Answer
A few games resolve this situation by dealing with it explicitly in the rules, and building the check system to accomodate how it handles this situation. The most notable one is Burning Wheel and its Let It Ride rule:
When in a situation like this, the success or failure of the attempt comes down to one roll. Players may cooperate or not, but once the dice are cast for the first time, the door will/will not be open and there can be no new roll by anyone for that goal until the situation significantly changes (like, they go fetch a battering ram, or they return a month later).
Lots of Burning Wheel's rules are tightly enmeshed in the rest of the rules, but this is one of the few that is easily separated and portable to other games. If your game of choice already has rules for assisting, you don't even need to houserule anything once you let your players know that you'll be following Let It Ride from now on. In the situation you describe, the fiction doesn't even need to change. "We all take turns bashing at the door. Eventually one of us must get through!" After establishing their method and their goal, they decide who's making the roll, everyone else adds bonuses for helping, and then the one roll is made.
This rule was built into Burning Wheel specifically because the author got the same feeling as you, that there's a problem with this common occurrence in games with skill systems. Most games test for the action, which under some circumstances virtually guarantees success (or failure) by just repeating the action. Instead of using task-based resolution like that for skills, Burning Wheel uses an intent-based resolution, where your goal is why you roll (and only once), but what you roll is determined by the method you use. Let It Ride is a key part of its intent-based resolution, in that it reminds everyone that they only get to do this once, so they need to bring all their resources to bear – or not, if someone in the party is opposed to the attempt. Either way, everyone has to commit to either pursuing this goal or not, before the roll happens. There's no hanging back to try after, because there's no second chance.
As an added bonus, it fixes a related problem, because the rule binds the GM too. The GM is forbidden from calling for multiple tests for the same task, so there's none of this sort of thing designed to railroad a failure: