It seems that the reason is that chemists don't care how many atoms there are in their reactions. It would be inconvenient: fractions of atoms don't make sense, and the numbers are far from human scale. They just want an agreement, so that even if they don't know the exact number of atoms in their test tube, they know they all have the same number. That's what units are for.
The definition of units are done in the most convenient and precise way possible. And they are updated as science progresses. For the kg, the artifact used to be the best we had, alternative methods weren't precise enough, they changed that to the Kibble balance because now, it is better. It may change again at a later time if we find something better.
Back to the mole, it used to be defined as the ratio between the kilogram and the mass of a 12C atom because it is the best we had. Now that we can count atoms with more precision, we decided to change it to just the Avogadro constant, which becomes fixed. Again it might change. What may happen (or may have happened, see Avogadro project) is that we can define the kilogram using the mole, making it fundamental.
The point of the mole unit is this: 1 mole of nucleons (protons or neutrons) has a mass of 1 gram. So while a mole is "just" a name for a large number (akin to billion or quintillion), the actual underlying exact count isn't very important. Instead what matters is making the stochiometric computations easy to do from first principles; given the formula of a molecule, and the atomic weights of each element, you can compute the mass of one mole, and dividing the mass of your stuff tells you how many moles without any other conversion factor. If we used a different unit of mass than grams, such as pounds, then it would make sense to use a similarly different definition of mole to avoid throwing extra units into the equation.
The difficulty with this simplistic view is that it turns out that nucleons actually differ in mass slightly, depending on which nucleus they're in (and, also, isotopic ratios matter), which means there are a few different definitions you can use for the number of nucleons in 1 gram of stuff. In practice, the difference is small enough that it doesn't matter for most uses, especially if you build it into your table of atomic weights.
You were also never going to test your scale with a weight that is stored in a vault in Paris. So far you would test with a replica of a replica of that weight, in the best case (getting a copy directly from the local standards body). With the new definition in principle anybody can create an object that weighs exactly one kg and sell it to you.
will this increase the probability of adoption of the SI system? I have often had the impression that some nations avoided transitioning to SI out of principle, not wanting to depend on the maintenance and cooperation of another nation?
It seems that the reason is that chemists don't care how many atoms there are in their reactions. It would be inconvenient: fractions of atoms don't make sense, and the numbers are far from human scale. They just want an agreement, so that even if they don't know the exact number of atoms in their test tube, they know they all have the same number. That's what units are for.
The definition of units are done in the most convenient and precise way possible. And they are updated as science progresses. For the kg, the artifact used to be the best we had, alternative methods weren't precise enough, they changed that to the Kibble balance because now, it is better. It may change again at a later time if we find something better.
Back to the mole, it used to be defined as the ratio between the kilogram and the mass of a 12C atom because it is the best we had. Now that we can count atoms with more precision, we decided to change it to just the Avogadro constant, which becomes fixed. Again it might change. What may happen (or may have happened, see Avogadro project) is that we can define the kilogram using the mole, making it fundamental.