  LEP precision on electroweak measurements was sufficient not to hamper the
extraction of Higgs couplings at the LHC. But the foreseen permille-level Higgs
measurements at future lepton colliders might suffer from parametric
electroweak uncertainties in the absence of a dedicated electroweak program. We
perform a joint, complete and consistent effective-field-theory analysis of
Higgs and electroweak processes. The full electroweak-sector dependence of the
$e^+e^- \to WW$ production process is notably accounted for, using
statistically optimal observables. Up-to-date HL-LHC projections are combined
with CEPC, FCC-ee, ILC and CLIC ones. For circular colliders, our results
demonstrate the importance of a new $Z$-pole program for the robust extraction
of Higgs couplings. At linear colliders, we show how exploiting multiple
polarizations and centre-of-mass energies is crucial to mitigate contaminations
from electroweak parameter uncertainties on the Higgs physics program. We also
investigate the potential of alternative electroweak measurements to compensate
for the lack of direct $Z$-pole run, considering for instance radiative return
to these energies. Conversely, we find that Higgs measurements at linear
colliders could improve our knowledge of the $Z$ couplings to electrons.
