Precision measurements are a powerful tool in the search for physics beyond the Standard Model, and their impact is often limited by our control of hadronic effects.  After a brief mention of the muon $g-2$ anomaly as an example of this challenge, I will focus on the closely related quantity $\Delta\alpha_{\mathrm{had}}^{(5)}(M_Z^2)$, the hadronic contribution to the running of the electromagnetic coupling. I will present a recent high-precision lattice QCD determination of the hadronic vacuum polarization in the Euclidean region, achieving sub-percent accuracy and confirming tensions at space-like virtualities with data-driven estimates based on $e^+ e^-$ cross sections. Using the Euclidean split technique, we obtain a precise determination of the running of $\alpha$ up to the $Z$-pole and compare it with global electroweak fits. I will conclude with an outlook on future improvements and the precision targets set by the FCC-ee program. This talk complements Tuesday’s Lattice Coffee Talk, where methodological details will be presented.