We know that the current symmetry of SM is $SU(2)_{fermions, left}*U(1)$ and the correct representation is the $2+2$ representation. I want to know what is the reason we don't consider the symmetry group to be $SU(2)_{lepton, left}*SU(2)_{quark, left}*U(1)$? Considering quarks and leptons are both represented in the fundamental representations.

I guess that it can lead to a fine-tuning problem like why the couplings to both sectors (leptons and quarks) are the same? (Surely if the couplings are the same then according to the unique Higgs vev the masses of bosons in both sectors turn out to be the same). Am I right?! Then to ease this problem doesn't one have to introduce another symmetry?!!

But is there any deeper reason that can really falsify this symmetry group as an alternative to the current one?

We know that the current symmetry of GSW is $SU(2)_{fermions, left}*U(1)$, and the correct representation of the $SU(2)_{fermions, left}$ is the $2+2$ representation. I want to know what is the reason we don't consider the symmetry group to be $SU(2)_{lepton, left}*SU(2)_{quark, left}*U(1)$? Considering quarks and leptons are both represented in the fundamental representations.

I guess that it can lead to a fine-tuning problem like why the couplings to both sectors (leptons and quarks) are the same? (Surely if the couplings are the same then according to the unique Higgs vev the masses of bosons in both sectors turn out to be the same). Am I right?! Then to ease this problem doesn't one have to introduce another symmetry?!!

But is there any deeper reason that can really falsify this symmetry group as an alternative to the current one?