Why do we take the quantization of forces so seriously even though the particles mediating the forces are not even real particles as per our theories? Recently I came across this question on this site, where the answer written by @AndrewSteane says  (paraphrased for convenience)

The electromagnetic interactions between two particles can be expressed as an integral  over all the ways in which one object interacts with the electromagnetic field which in turn can interact with the other object. These interactions can be expressed in a number of ways, but a particularly nice way is to assert that an electron emits something called a virtual photon. This virtual photon is different from a real photon in that it should not be considered as a thing which could in any sense go on its way to the rest of the world and interact with anything else. Rather it is a way of talking about part of the interaction between the particular two electrons under consideration.

And hence my question: If they aren't even real photons, why did we call them, well, photons? And then why is the quantisation of all the $4$ forces taken so seriously when it is just for mathematical elegance and nothing else?
 A: You ask:

If they aren't even real photons, why did we call them, well, photons?

Virtual particles belong to the calculations for particle interactions using Feynman diagrams.We call them virtual photons (or electrons or ... as the case may be) because they are described by a special relativity four vector that has all the attributes of the named particle except the mass, the mass of a virtual photon is a variable within the calculation.

And then why is the quantization of all the 4 forces taken so seriously when it is just for mathematical elegance and nothing else?

Physics uses mathematics not for elegance, but in order to calculate  numbers , as are the crossections for interactions. The models based on quantization fit the data and are predictive of new data and observations. The language of physics is mathematics, not words. Words are shorthand for mathematical concepts but to really understand what physics is about needs mathematical knowledge.
A: Virtual particles are a useful calculational tool in pertubation theory, and provide an easy "conceptual hook" for popular science articles (and even for some purposes in physics). But it's dangerous to attach too much meaning to a picture of little points zipping around space. The full picture is much more complicated: in the standard model the fundamental objects are quantum fields, and those are the things that are interacting. Particles are excitations of the fields. Any attempt to provide a "mental picture" of quantum fields with words is bound to fail in some respects, because they are things very unlike the macroscopic objects we encounter in everyday life. What's unambiguous is the mathematics of quantum field theory, but this does require some intense study to master.
The quantization of fields is taken seriously because it provides many predictions that have been verified, and explains phenomena like radioactive decay that are otherwise inexplicable. It also does away with "wave particle duality": there are only fields, and "particles" are the quantum units of the fields.
