I will use this excellent comment as a starting point:
[Nicolau Saker Neto] I think it's possible to argue that metallicity (or more generally, delocalized bonding) is more natural, and it's the non-metals which are unusual - why do we have a bunch of those?
Definition and structure of metals
Mostly, we think of metals as solids, although the liquid mercury is commonly defined as metal as well. While the OP's question is about elements, alloys are classified as metals as well, even when they contain some non-metalls like the carbon in steel.
So to be considered a metal, an element has to be in a condensed physical state. In a solid or liquid, the atoms will be close together; in an element, they will all be the same. Either the atoms will interact with each other in the same way (like in a metal), or they will form groups of strongly interacting atoms (called molecules) interacting less strongly with other groups (like liquid bromine or solid iodine). The former seems more "natural", given that the atoms are all of the same type. The absence of molecules and the abundance of nearest neighbors gives rise to the band structure of metals (and the delocalisation of electrons) that explains many of their properties.
Temperature and pressure
[OP] How on earth is it possible for 92 of the 118 discovered elements to be metals!?
On earth, indeed.
If an element is in the gas phase, we typically don't describe it as a metal. This includes hydrogen, oxygen, nitrogen, the halogens up to chlorine and the noble gases up to radon. Some of these elements (including hydrogen, oxygen, nitrogen, chlorine, and xenon) solidify and form metallic phases when under sufficient pressure. A paper on high-pressure forms of chlorine claims [2]
Under sufficient compression, all molecular systems are expected to collapse into close-packed metals.
On the other hand, if we increase the temperature, most metals will vaporize (or even form a plasma). This shows that the number of elements we classify as metals depends on the ambient conditions we experience on the surface of our planet these days.
There are also elements that have allotropes, with one metallic and one non-metallic. A good example is tin, which has a metallic allotrope (white tin) and a non-metallic allotrope (gray tin), which is a powder and does not conduct electricity. Carbon comes in the form of diamond (non-conducting) and graphit (conducting). Boron has metal-like allotropes as well. Finally, metalloids are semiconductors, so their metallic character depends on temperature.
This answer focused on physical properties. The key chemical property of metals is that they tend to form (monoatomic) cations. Taking away electrons from an isolated atom always costs energy (all ionization energies are positive). Interacting with a polar solvent (like water, made of nonmetals) or with anions (made of or at least containing nonmetals) offsets this cost in some cases, so we observe the formation of cations. This means that without non-metals, we would not be able to observed metals forming cations. To explain (or rationalize) which elements form cations or anions, we invoke the electronic structure of atoms, as you will find in any textbook. The introductory textbooks focus on main group metals and non-metals, yet the bulk of metals are transition metals and f-group metals, where things quickly become hand-waving (OP's "jumping" electrons).
Cause and effect
[OP] Specifically, what is the exact physical cause behind this phenomenon?
Sorry, other than pointing to experimental evidence showing that most elements have metallic properties under ambient conditions, and adding that this is well-described by theory (but the theory is complicated and the applications are subtle), I got nothing.
- Dalladay-Simpson, P.; Binns, J.; Peña-Alvarez, M.; Donnelly, M.-E.; Greenberg, E.; Prakapenka, V.; Chen, X.-J.; Gregoryanz, E.; Howie, R. T. Band Gap Closure, Incommensurability and Molecular Dissociation of Dense Chlorine. Nat Commun 2019, 10 (1), 1134. DOI:10.1038/s41467-019-09108-x.