What is the search incident to arrest exception to the warrant requirement?

Imagine if when police arrested people, they just slapped handcuffs on them and put them in the police car as-is. What if they had a weapon on them? What if right before they were arrested, they hid evidence under the cushion of the seat they were sitting on? There's a lot of safety and logical reasons to support a search incident to arrest, but at the core it is as simple as "if someone is being legally arrested, then police are allowed to search them and the immediate area around wherever they were when they got arrested in case they stashed stuff when they saw police coming."

For this type of search to be lawful, there are a few restrictions:

(1) The arrest itself must be lawful, meaning there must be probable cause or a warrant to make the arrest. In other words, if a police officer randomly arrests a person because they looked at the officer funny, yes the officer technically arrested them, but the search incident to arrest will not be valid since the arrest was not lawful, which means any evidence discovered as a result will be inadmissible under the exclusionary rule.
(2) The whole purpose of the search incident to arrest exception to the warrant requirement is to (a) provide extra safety for officers by enabling them to make sure a weapon isn't accessible to the person they are arresting, and (b) to try to preserve evidence related to the arrest they are making. With this in mind, the search that is being performed as a result of the arrest must be done contemporaneously in time in place, which means the "free pass" police get under this exception has to be used at the same time and at the same place the arrest is being made. For example, imagine if police have probable cause to arrest Bob inside of his home. When they open the door, they find Bob sitting on top of a couch. When they arrest him, they are allowed to also search the couch (and the area immediately surrounding the couch) but they need to do so at the time of arrest. They can't come back the next day and bust down the door to search the couch because they forgot; at that point, they would need a warrant because the search is no longer happening at the same time as the arrest.
(3) Let's stick with the previous example from (2). When police arrest Bob sitting on his couch, they are allowed to search the couch because it is within his wingspan. In other words, it is possible that when Bob heard the cops entering his home, he hid evidence. Where could he likely have hid evidence quickly? Anywhere within reach. Thus, anywhere within reach can be searched: the couch, maybe the coffee table next to the couch, maybe a small box located on the coffee table. All of the places within the wingspan are arguably fair game to be searched under this exception. In contrast, police cannot wander around the house and search anywhere they want.

When it comes to arrests involving drivers or passengers of cars, police are allowed to search the inside of the car in two situations:
(1) When the officer cannot secure all of the suspects they are arresting. For example, imagine if an officer saw a car drive by with 5 teenagers all smoking weed. When the officer pulls the car over and orders the 5 teenagers out, he realizes he only has 1 set of handcuffs. At this point, the officer is allowed to tell the teens to stand back while he searches the car. Why? Because it is possible that one of the unsecured teens could run back to the car and destroy evidence of the crime or retrieve a weapon. In contrast, if all arrestees are secured, then a cop cannot search the vehicle unless the second situations applies:
(2) If the officer has reason to believe there is evidence inside of the vehicle related to the crime the defendant is being arrested for. In other words, if you're arresting a driver because you believe they were smoking weed, you can search the car for drugs. In contrast, if you've arrested a driver because of a suspended license, then you're not allowed to go searching their car for more crimes without first getting a warrant.