Why don't we have a theory solely of weak interaction, like QED or QCD? I.e $SU(2)$ gauge theory describing neutrinos and massive $W^{+-}$ bosons.
But instead we have a unified electroweak theory under $SU(2)\times U(1)$ gauge symmetry involving symmetry breaking by Higgs field. What is the reason for this? Is it something related to problem of giving a mass to chiral fermions?
You are asking three questions, really:
The fitting of these three points is the quasi-magic of the solution to the SM puzzle.
The first was solved by Glashow in 1961. The Feynman—Gell-Mann theory of the charged weak interactions relied on the two left-chiral charges $$ 2T_+=\int d^3x ~~\nu_e^\dagger (1-\gamma^5) e, \qquad T_-= T_+^\dagger \\ \leadsto ~~~ [T_+,T_-]= \int d^3x ~~~\left ( \nu_e^\dagger (1-\gamma^5) \nu_e -e^\dagger (1-\gamma^5) e \right )/2. $$ But this is not orthogonal to the vector EM charge $Q=\int d^3x ~~ e^\dagger (1-\gamma^5) e $ : it contains it, linearly combined together with an improbable ("cockeyed") neutral-current charge, of spectacularly counterintuitive chirality; unknown back then, and only discovered after a decade. So the weak interactions are inseparable from electromagnetism.
The mass term for the electron, $m_3 \bar e ~ e$, is not invariant under the action of the chiral $T_{\pm}$. Introducing the Higgs and SSB of the global symmetry magnificently solves that problem.
The 4-Fermi interaction suggests massive intermediate vector bosons. Renormalizability/computability all but dictates that the vector bosons be gauge bosons made massive by the Higgs mechanism. Note this is the iffiest part of the puzzle set by Weinberg and solved by 't Hooft and Veltman...
Without the Higgs mechanism we can't have a gauge theory with massive bosons, the reason is very simple, let's take the example of why we can't have a massive electromagnetic photon:
Electromagnetism is described by the group $U(1)$, this means that under a gauge transformation our vector boson $A_{\mu}$ transforms as:
$$A_{\mu} \rightarrow A_{\mu} + \partial_{\mu}f$$
Mass terms for bosons are of the form : $m A_{\mu} A^{\mu}$, but we have that under a gauge transformation this transforms as:
$m A_{\mu} A^{\mu}\rightarrow m A_{\mu} A^{\mu} + 2m A_{\mu}\partial^{\mu}f+ m \partial_{\mu}f\partial^{\mu}f$, i.e. it's not gauge invariant.
For more complex groups, like $SU(2)$, the reason is exactly the same: the mass term is not gauge invariant.
In conclusion it's not possible to have a gauge theory where the vector bosons are massive, because it would break the gauge invariance, for this reason it was not possible to formulate a quantum theory of the weak interaction without the Higgs Mechanism, because since this is a "contact" interaction, we already knew that it would have been mediated by massive boson.
