Neutrons turn spontaneously into protons, which is called beta decay, and which happens in any nucleus with too many neutrons. This includes the free neutrons, which decay into protons in minutes.

Neutrons and protons differ in their composition, a neutron being made of 2 d quarks + 1 u quark, while a proton is made of 1 d quark + 2 u quarks, much in the same way as a nucleus of tritium differs from a nucleus of helium 3, the former being made of 2 neutrons + 1 proton, while the latter is made of 1 neutron + 2 protons.

For the strong interactions, nucleons (i.e. protons and neutrons) and nuclei are analogous to what atoms and molecules are for the electromagnetic interaction.

The strong interaction attempts to neutralize the hadronic charge (a.k.a. color charge), while the electromagnetic interaction attempts to neutralize the electric charge.

To a first approximation, the hadronic charge is neutralized in nucleons and the electric charge is neutralized in atoms.

However, because of the movement of the quarks inside of a nucleon and of the electrons inside an atom, the neutralization of the charge is imperfect and there remain some residual forces of attraction, respectively strong and electromagnetic, which bind the nucleons into nuclei and the atoms into molecules. Because they are just residual forces, the binding forces between nucleons in a nucleus are much weaker than those between quarks in a nucleon, similarly to how the binding forces between atoms in a molecule are much weaker than those that bind most of the electrons to the nucleus in an atom.

Do you think it’s possible that the periodic table is too simple of an abstraction and that quarkish elements exist which cannot be aligned on the table but perhaps are never seen in nature or extremely rare?

The so-called elementary particles, which are divided into leptons and hadrons, are all "achromatic", i.e. the color charge is null for the whole particle.

While the leptons may be considered as truly elementary, at least in the current state of knowledge, the hadrons are composed of quarks, and the quarks have non-null color charge.

At present there is no hope of being able to produce any particle where quarks are separated, i.e. any particle with non-null total color charge, because when the distance between quarks increases the attraction force between them also increases (like they would have been bound by an elastic spring), until the force becomes high enough so that a pair quark-antiquark is generated, so the original hadron may split into 2 hadrons, both of which have null color charge and no free quarks can be produced (e.g. the quark initially being pulled apart is split away, but it takes with it the antiquark newly generated, forming a meson particle instead of a free quark).

Attempting to separate the quarks of a hadron has a result somewhat analogous to the attempt of separating the north and south poles of a magnet, when breaking the magnet produces a new pair of north and south poles, so you get 2 new magnets, each with a north and a south pole, instead of getting a north pole separated from the south pole.

Therefore, because neither free quarks nor combinations of quarks where the color charge is non-null can be produced, no "quarkish" elements can exist.

Nevertheless, while the normal chemical elements have nuclei composed of nucleons, i.e. protons and neutrons, it is possible to have nuclei composed of other hadrons, i.e. nuclei where besides protons or neutrons there are one or more of the so-called hyperons, which have a similar structure to nucleons, but which contain some heavier quarks than the u and d quarks that compose nucleons (there are also extremely short-lived heavier hadrons that contain more than 3 quarks, as long as the total color charge is null).

However, all hyperons have an extremely short half-life, much shorter than a second, so if such an exotic element containing hyperons in its nucleus were formed due to a very unlikely sequence of collisions between particles with very high energy, it would decay extremely quickly.

At the huge scale of the Universe, even extremely unlikely events may happen somewhere, so perhaps a few atoms of such hyperonic chemical elements have a transient existence somewhere (during a small fraction of a second), but their quantity must be truly negligible.

While a few atoms of such elements can be produced artificially or naturally, there is no chance to ever produce a quantity great enough to make a piece of material that you could see with your eyes, much less take in your hand (ignoring the extreme radioactivity of such an element, which would destroy anything close to it).

The only possible exception might be in extremely high gravitational fields, i.e. inside neutron stars and black holes, where there may be a chance that such hyperons could become stable due to the extreme pressure, but we do not really know the possible structure of matter in such conditions and in any case at such pressures there would be no chemical elements in the normal sense, as there would be no free electrons.