Can quarks be considered real and elementary? In our current theories all hadrons are made up of quarks and gluons.
This view reduces considerably the big family of hadrons by providing a very logical structure in which all quantum properties values of the hadrons are originated from the valence quarks' quantum properties. But since they cannot be observed isolated, in the sense that electrons or positrons can be, the theory assumes color confinement is just part of the game.
However actual calculations of QCD use a view where they are quarks and gluons are distributed in a space-time volume, which is needed to describe much of the rich phenomena of high energy physics.
But all of this makes me wonder: 
Are quarks real and elementary? I mean, is there more evidence supporting that they are more than just a model that works?
Since hadronic matter cannot by split into pieces smaller than the smallest hadron, couldn't reality be explained also by a model where all observable hadrons are the elementary ones, which follow rules for transforming into other ones, etc.?
By the way, this last idea was already presented by Hagedorn on his very famous known paper, in the end where he tries to give a phylosofical point of view to his model, so I wonder why was it discarded.
 A: What you propose is called nuclear democracy, and was very popular in the days before the standard model emerged. See also the discussion in http://www.physicsoverflow.org/22971 , where you can read about (among others) my present view of it.
A: As mentioned in the OP quarks vastly simplify the theory of hadrons, like atoms did chemistry, and despite confinement Rutherford-like experiments were performed for them too, by Friedman, Kendall and  Taylor who received the Nobel prize for it in 1990: "unexpectedly large numbers of electrons being scattered at large angles provided clear evidence for the pointlike constituents within nucleons. These constituents are now understood to be quarks."
But can quarks still be considered non-existent? Technically, yes.  This is the answer Mach gave about atoms in the 19th century: they are just fictions, and the theory can be re-arranged in a way that eliminates them, e.g. by connecting only measurable quantities to each other, and purely mathematically (Mach and a leading chemist Ostwald even refused to mention atoms in their works after 1870). This remained true even after Rutherford's experiments, and it remains true today despite the technology that (ostensibly) "allows them to be imaged, split and smashed". Indeed, one can even eliminate everyday objects and reduce everything to sensations, as some positivists suggested. But doing so will make for a very unattractive theory. 
Of course, it can also go the other way: at the end of 19th century the ether was a solid element of reality. Some even expected a theory of everything out of it, like Michelson in 1902, see Kragh's Quantum Generations, p. 4:

"The day seems not far distant when the converging lines from many apparently remote regions of thought will meet... Then the nature of the atoms, and the forces called into play in their chemical union... the explanation of cohesion, elasticity, and gravitation — all these will be marshaled into a single compact and consistent body of scientific knowledge... one of the grandest generalizations of modern science ... that all the phenomena of the physical universe are only different manifestations of the various modes of motion of one all-pervading substance — the ether." 

And then the ether was no more. But that does not happen very often.
A: If it conserves like a quark, and scatters like a quark, then it is a quark.
A: "unexpectedly large numbers of electrons being scattered at large angles provided clear evidence for the pointlike constituents within nucleons. These constituents are now understood to be quarks."
Exact formula - do not prove that quarks exists, but pointlike constituens within nucleons, which undertood now as quarks. Logically perfect :)
To my personal view, the question of elementary / composite particles is most important, and idea of nuclear democracy was the root of problem. 
Nuclear democracy suggested that all particles in Particle Zoo are elementary, that means that they did not consist of each other. That is normal view for physisist of that time and very strange view for chemist. Physisist tends to imagine particles as points, and chemist tends to imagine particles as elements, compounds of elements and reactions. Quite different backgound - physisists of that days had no expecience in classification of particles and reactions and chemists had such experience for the centuries, as it is a key in chemistry.
That nuclear democracy was already taken into account into the Standard Model to some extent, so no reason to come back again. But nuclear democracy needs strong experimental proofs for chemists, although it is natural for phisicists. 
For that reason, I think, big class of theories was not even checked, and it is similar to your question. Yes, other approaches can exist. But never were checked.
For example, Rutherford, who predicted the neutron, suggested that neutron is similar to nucleus composite particle, which consist of proton and negative particle. He suggested also, that that particle can be "inner" electron of the nucleus.
It became obvious that electron cannot be inside the nucleus. But what about main suggestion that neutron is composite (compound), which consist of proton and negative particle (other then electron)? Nobody even checked the particles of Particle Zoo for possibility to be a partner for proton in nuclear binding. 
Instead of this, all particles were suggested equally elementary, its classification were made with this unprooven nuclear democracy idea, symmetry was applied - and we have natural result - the quarks model.
