Your question concerns the D & D 5E capability verification system a bit backwards. The mechanic is indeed called ability Checks, not skill Checks, and it should be used to first consider the ability and then find out if the competence could apply in a skill. From the Skills Checker section in Chapter 7 (my emphasis was added):
For each skill check, the DM decides which of the six skills is relevant to the task at hand and the level of difficulty of the task, represented by a difficulty class.
And later in this chapter, in the Skills section:
Sometimes the DM asks for a skill check using a particular skill, e.g. B. "Wisdom (Perception) Examination". In other cases, a player asks the DM if a particular skill is suitable for a test. In both cases, proficiency in a skill means that a person can add their competence bonus to ability assessments involving that skill. Without knowledge of the skill, the person performs a normal skill test.
So instead of looking at it from the perspective of "How do I make a check on cooking utensils?" To look at it, we have to look at it from the perspective of "Why does the DM require a capability check?" Consider. There are many things that are about cooking utensils that are unlikely to need to be checked, since a normal everyday meal in a household where all the ingredients are readily available usually does not pose a great risk of failure or much influence success. But sometimes the character tries to do something that makes sense to use a roll to see if it succeeds, so the DM requires a skill check. And if knowledge in using cooking utensils were helpful in achieving the goal, the competence bonus would be added. Here are some skill checks I can imagine where you could sensibly add the skill bonus for cooking utensils:
- A strength check to scrape fried foods from a pan. (This is admittedly a bit boring, but maybe if there was some time pressure to brush something to make a nobleman happy or something.)
- A skill check to do something with fine details, such as decorating a cake.
- A constitutional review to make foods that are needed for a long time in a hot, muggy, steamy kitchen.
- A secret service check to remember an old family recipe.
- A wisdom check to see how someone else makes a recipe to copy it, or to decipher what kind of food he likes.
- A charisma check to prepare food in an appealing way or to make friends with food.
As you've noted, apart from the general suggestions in the section "Using every skill," it's all about the DM, when skill checks are required, and what ability is available for the task the character is trying to accomplish. Different DMs definitely have different philosophies and approaches. And sometimes really more than one skill can be meaningfully applied. I have certainly called for charisma checks (as a DM) to ensure that either their belief or cookware skills are used, depending on what they had as they tried to make friends with food. However, once the DM determines what ability to use for skill testing to determine what the character is trying to do, a throw is required. In general, it is not that difficult to find out if the skills of a tool are correct for the litter.