Why do all fields in a QFT transform like *irreducible* representations of some group?

Gell-Mann's totalitarian principle provides one possible answer. If a physical system is invariant under a symmetry group $G$ then everything not forbidden by $G$-symmetry is compulsory! This means that interaction terms that treat irreducible parts of a reducible field representation differently are allowed and generically expected. This in turn means that we will instead reclassify/perceive any reducible field in terms of their irreducible constituents.

This is only semantics. A reducible representation $\mathbf R$ of the symmetry group can be decomposed into a direct sum $\mathbf R_1 \oplus \cdots \oplus \mathbf R_N$ of irreducible representations. A field that transforms as $\mathbf R$ is the same thing as $N$ fields, which transform as $\mathbf R_1, \dots, \mathbf R_N$. When talking about fundamental fields, we can therefore assume that they transform as irreducible representations of the symmetry group.

Irreducible representations are always determined by some numbers, labeling the representation, which correspond to the eigenvalues of some observables which are invariant under the (unitary) action of the Lie group.

If the group represents physical transformations connecting different reference frames (Lorentz, Poincare',...), these numbers are therefore viewed as observables which do not depend on the reference frame so that they define some intrinsic property of the elementary physical system one is considering.

If the group represents gauge transformations, these numbers correspond to quantities which are gauge invariant. In this sense they are physical quantities.

Finally, it turns out that in many cases (always if the Lie group is compact), generic unitary representations are constructed as direct sums of irreducible representations. This mathematical fact reflects the physical idea that physical objects are made of elementary physical objects (described by irreducible representations)