Stupidity: Project Pegasus is a billion dollar structure with no security guards besides the couple dudes Fury shows the badge to at the entrance, no video surveillance, and once SHIELD arrives nobody has to even open a locked door anymore.
Suggested correction: First of all, Fury was allowed in as it's a government facility and he works for the government and as a SHIELD agent is allowed access. It's inside a mountain and they passed multiple security guards as they drove in, armed guards. Everything is thumb prints and cameras which was quite elaborate for that time. To say they lack in security is quite an understatement. SHIELD has quite some authority and can easily take control in the Pegasus project facility.
You are right about the main entrance being truly secure both for guards and strategic position. The problem is that they are free to just roam the facility for an hour, blast through doors, not a soul in sight, no evidence of camera monitoring the inside of the structure, and once the gag of the pad is finished, no door requires it. Actually, funnily enough you can see a guard of the place opening the elevator for Fury and the supervisor, as if the thumprint scan was needed to even get into the elevator itself, but Keller then just walks into the archive just fine (from a different door than the one Vers blasted). There are keypads to exit places (for instance the hangar, when the agents in pursuit break through the door you can see a keypad on the wall) but only when it's convenient (Vers and Fury walked through that same door with no problem, not to mention the fact that the whole stairs seem to have none, which is funny for a place that has keypads both sides of doors).
Stupidity: The bad guy who spies on Bosley is visible from the very beginning of the scene, when Elena is not even inside the cafe yet. The cafe does not have many customers and the bad guy is using a loud vintage typewriter with a big mirror mounted on top (!). The Angels are portrayed as being super smart and their setup is so careful, but somehow they managed to miss something amazingly obvious. Conversely, nobody would have ever paid attention to a laptop or any silent, modern-age device perhaps with a camera or something similar, so the bad guy chose the most blatantly conspicuous accoutrement for his spying job. Try showing up to a cafe punching the keys of a big clunky typewriter and literally everyone will be looking at you. (00:18:25)
Stupidity: Despite the fact that the villains know exactly where Brogan is for the better part of the movie thanks to a tracer, and that they display constantly that they are able and willing to hit with bold military operations even to the point of hitting an urban area overruling law enforcement (apparently Clay has authority to do that too), Brogan is able to get undisturbed back in the US through his friend's plane. Considering he was officially marked as a rogue agent and killed 8 DIA operatives, it's hard to see how Baron's plane could ever land on American soil after a long intercontinental flight.
Stupidity: Thor looked for (and found) "weapons" in the back of his parents' closet, but didn't think of getting knives from the kitchen. Also, when Claude was in the house, Thor put the paintball gun in Max's backpack in case someone messed with them when going for the drone but had not thought of using it before Claude arrived.
Stupidity: The marble somehow got under Don's skin at the bottom of his leg and was moving up the length of his body. Instead of heading off the marble in order to intercept and stop it from continuing up his leg, Don got a knife/box cutter and jabbed it into his skin near the bottom of his leg and trailed the marble, causing a long gash that served no purpose (other than creating a lot of blood and making a gory scene). The marble eventually exited his body via his eye. (01:04:40)
Stupidity: When Joan is in hell and is confronted by the overlord, she attempts to shoot him with the BFG-9000, so the overlord uses his powers to telekinetically blast Joan away and disarm her. A minute later, Joan picks up the BFG-9000 again, but the overlord never considers using its powers to blast Joan away from him again, which results in him getting killed.
Stupidity: The kidnappers steal a silver Citroen van and take it to a junkyard to switch the plates. They steal plates off an older green color Asian type sedan and place them on the van. If the cops run the plates as they normally do random checks as they drive, those plates will not match that vehicle. They should have found the same make and color of Citroen van they stole and swapped plates for plates, the chances of the owner noticing are small. Now if the plates get run they match that vehicle. (00:00:38 - 00:01:18)