  We present systematic study of isospin impurities ($\alpha_{\rm ISB}$) to the
wave functions of $T=1/2$, $11\leq A \leq 47$ mirror nuclei and the
isospin-symmetry-breaking (ISB) corrections ($\delta_{\rm ISB}^{\rm V}$) to
their ground state vector $\beta$-decays using, for the first time,
multi-reference charge-dependent density functional theory (MR-DFT) that
includes strong-force-rooted class-III interaction adjusted to correct for the
Nolen-Schiffer anomaly in nuclear masses. We demonstrate that, unexpectedly,
the strong-force-rooted isovector force gives rise to a large systematic
increase of $\alpha_{\rm ISB}$ and $\delta_{\rm ISB}^{\rm V}$ as compared to
the results obtained within MR-DFT that uses Coulomb interaction as the only
source of ISB. This, in turn, increases a central value of the $V_{\rm ud}$
element of the CKM matrix extracted from the $T=1/2$ mirrors bringing it closer
to the value obtained form the purely vector superallowed $0^+ \to 0^+$
transitions. In order to compute the value of $V_{\rm ud}$, we performed
precision calculation of the Fermi matrix elements in $A=19, 21, 35$, and 37
mirror nuclei using DFT-rooted configuration-interaction model that includes
all relevant axially-deformed particle-hole configurations built upon Nilsson
orbitals originating from the spherical $sd$ shell. Our calculations yield
$|V_{\rm ud}|=0.9736(16)$.
