According to the bond triangle, compounds don't exist as solely ionic or solely covalent, but rather have ionic, covalent, and metallic character to them. So each bond type is connected and similar in a way. However, when learning about the different structures each bond type forms as a solid, they seem to form vastly different structures. Ionic compounds form lattice structures, covalents have molecular and network structures, and metallics have metallic lattices. Why is this so? For example, why don't ionic compounds form molecules if they contain covalent character? Is it because there are sort of transition structures that are mixes of the different structures mentioned above?
Premise: in spite of my comment, which I leave posted for reference, I've decided that I was overthinking about the reasons that lead OP pose the question and that is better to answer to what is certainly asked.
However, after having tentatively edited the title to reflect what the text deals with, I must amend the question at two points:
1 - "compounds don't exist as solely ionic or solely covalent..." the subject should rather be bonds instead of compounds. Also the useful mantra "there are not fully ionic or fully covalent bonds" shall be used with more parsimony or at least more carefully, as it is true context given (phenomenologically it does not holds in biatomic hydrogen or for the C-C bond in ethane, etc...)***;
2 - about the various structures and hierarchies listed by OP, what is relevant is that, when solid bulk materials are considered, ionic bonds leads to ionic lattices while covalent bonds can lead to both covalent lattices or to discrete molecules afterwards kept togheter by various forces which aren't strong enough nor have the nature of the aforementioned bonds.
Answer We shall take as starting points that
1 - an element faced to other atoms might react to achieve an overall lower energy.
2 - Generally enough to be used here, atoms tend to fulfil the octet rule.
3 - The way they do so depends on their absolute electronegativity as well as on the electronegativity difference between them.
(Of course what determines electronegativity and why atoms with a filled octet are particularly stable can be the subject for two or more questions).
When the difference in electronegativity is high, typically between elements at the extremes of the periodic table periods, the octet is filled by a true electron transfer. In the simplest case, this leads to a couple of opposite charged ions (real and relatively stable particles, as far electric neutrality is satisfied they can individually exist in the proper conditions, for instance as solvated species in water) and the overall energy is further lowered by their electrostatic pairing.
Basically, an ion can be treated as a point charge with a spherical electrical field. There is no need to locate the extra electron of an elemental anion, or to know where the electron leaved a "hole" in the cation counterpart. The two are attracted by the coulombic force and as such direction trivially matters only in the minimal entity they can form, kind of a molecule corresponding to the minimal formula.
An extented bulk made of ions just obey to electric neutrality and geometrical constraints that are somewhat "external", kinds of possible space-filling arrangements similar to those of a number of balls.
Indeed, in common conditions, ionic bonds hold togheter ionic crystals rather than molecules.
Conversely, when the difference in electronegativity isn't that high and when fulfilling the octet would require a transfer of too much electrons, the latter are instead shared in between the nuclei in a region of space that is indeed the bonding orbital. Although bonds are ultimately of electrical nature, the situation now does not involve electrostatic charges with their spherical fields, but defined directions as soon the involved element combines with multiple atoms.
Most molecules satisfy the octet rule within themselves, and the geometry constraints (VSEPR suffices, for this discussion) is what gives their shape.
The molecules, if eventually lead to a solid, arrange in crystals or amorphous bulks hold by various forces such as H-bond, permanent dipole interactions, or what often goes under the generic umbrella terms dispersion forces or Van der Walls forces. No need to go into detail, here. Those aren't chemical bond on the base of their strength and/or different nature. It is also quite matter of nomenclature and is usually easier to see why they are treated apart rather than writing down. The bulk so formed is called a molecular solid, and its arrangement its dictated by the shape and packing of the molecules and the predominant forces between them. It can even be that molecules form dimers, eg by means of H-bonds, and these dimers then pack via dispersive forces; etc.
When covalent bonds are involved there is another possible outcome, of which C diamond is the typical example. In that the geometry of the atoms allows for the building of a truly 3-D lattice. Note that - in the context of this thread - even this case is basically as the above one, just the bulk isn't a molecular solid but can be seen as a very very big 3-D macromolecule. It is called a covalent solid.
Finally, when we say that
bonds aren't solely ionic or solely covalent
we mean that, unless the molecule is highly symmetric about the bond under discussion, electron shared in a covalent bond aren't equally distributed between the bound atoms. In other words, the electron density is higher towards the element - or the molecular moiety - having the higher electronegativity.
As such, most of the time "there are not solely covalent bonds" means that in reality the vast majority of covalent bonds are polar, ie there is a net electrical moment along the bond direction, the intensity of which depends on the electronegativity difference between the bound atoms/moieties.
For metals, I would say that the pictorial description of cations immersed in a sea of electrons can be usefully recalled. It suffices to explain while metal bonds are not directional and again atoms in a solid metal are arranged to fill the space as spheres do - with the addition that being those spheres positively charges, the interstitial volume is filled by a kind of "continuous" negative fluid, ie the electrons spherically distribute all around each atom and as such ultimately mobile through the bulk.
Leaving the alkali metals, as elements valence electrons are on less symmetrical orbital, increases the covalent bond character leading to less simple structure and specific direction between planes. Still, the bond is responsible for holding the whole bulk.
***at another level, which however has no phenomenological relevance, even a completely apolar covalent bond, ie one with no electrical dipole moment, can be thought as a hybrid of resonance of which some limiting forms have ionic character. In such a sense it might be regarded as supporting the idea that all covalent bonds have some ionic character. However, the true meaning is that we can regard A2 as a hybrid of the limiting forms A-A, A+ A-, and A- A+. The weight of the latter is certainly small if not negligible and especially has not direct consequences on the way A2 molecules build up their solid phase. In a less symmetric case, AB (A-B; A+ B-), an electrical dipole moment would emerge. Still the whole entity would be neutral seen from a sufficient distance (that of the adiacent molecules) and as such there is no reason for an ionic lattice to form. At most the permanent dipole would dictate or influence how the molecules arrange in a molecular solid, when they do.