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This list isn’t going to work for every author, reader, or story.  Going to try to be as general as possible, but we’ll see how it goes.

1. If you want to remove a basic skill from a character halfway through a story for ‘flavor’ then you probably shouldn’t do it.  If you’re going to take away something that all other characters know from one then you should do it from the beginning.  Sure, it could be a shock that a hero can’t read or write, but bringing it up late in the story and out of nowhere should require a special edit.  For greater impact, you need them to avoid situations that would reveal this secret early.  Otherwise, it does come off as a random choice.
2. Even though it isn’t really basic, cooking is not a female-only skill.  So, a male character cooking shouldn’t be surprising solely because of his gender.  He shouldn’t be standing there as an anomaly since most people learn the basics.  For that matter, not every woman knows how to cook, sew, clean, or possess other ‘housework’ skills.  That isn’t genetic.  Much of these are basic skills that everyone needs to learn.
3. A character who doesn’t know a basic skill shouldn’t gain it within a few days.  The reason these skills are considered basic is because nearly every adult has them.  They were taught when the adults were children, who tend to be curious, interested in learning, and no shame in lacking the skill.  This is why it is harder to teach an adult these things than a child.  So, if you create this as an obstacle alongside a greater story, they are going to be distracted and limited in progressing.  Nobody learns to read overnight.
4. A skill that requires advanced training isn’t something that every character will have unless you establish it as being taught from a young age.  A town of hunters would be skilled as they teach their children.  Expanding this to the entire world where people live in different biomes and civilizations doesn’t work.  This means you need to consider if such a skill should be unique or general.  If the latter, you will need to explain how everyone knows it.
5. Just because a person can’t read or write, it doesn’t mean they talk like an uneducated caveman.
6. Yes, it’s possible for someone to be a natural.  One could even argue that a protagonist should be this since they are the most important character.  They are seen as special and more advanced than those around them, which is why they have the spotlight.  All of that being said, doing this all the time creates a boring, unchallenged hero who nobody can see failing.  Even if they do fail, the audience will find it unbelievable since they’ve already been shown to be perfect.  Giving them ‘basic’ or more advanced skills that they struggle or are unable to learn would help avoid this problem.
7. There are different categories of basic skills that one should consider.  You have those needed for basic living such as reading and writing.  Then, you have those that are general career based such as hygiene, organization, and maybe driving.  After that, you have job/role specific skills that one still has to learn for their purpose.  This last one can include carpentry, hunting, magic, swordsmanship, piracy, and the list keeps going.  The point is that on top of basic skills needed for a person to simply survive, there are basic skills for their roles as well.  Just makes this whole seem much more complicated.