Several misconceptions undermine this reasoning.
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The ionization energy puts an electron in vacuum, but conduction electrons are always near a nucleus in a metal, or rather near many nuclei. The energy differs a lot. In fact, the electrons' energy is much lower in a piece of metal than in individual atoms. That's why atoms bind to form a solid.
Consistently, the same element can become a metallic solid just by pressure. Solid hydrogen does it. Its ionization energy didn't change, but the energies accessible to the electrons in the solid change with the density.
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Bonds are not local to two atoms. The valence electrons delocalise over a complete molecule, where "molecule" means the macroscopic piece of matter for a metal or a ceramic. This holds for insulating crystals too. At organic compounds, sigma electrons too extend over a complete molecule.
Though, explanations and reasoning, including reaction mechanisms, are done as if electrons were local to one atom or two, which was the state of science many decades ago. This seems very efficient at predicting and designing reactions, hence is useful and necessary, but it's wrong, and has misled you. I wish someone some day makes a usable theory of bonds that is consistent with the rest of human knowledge.
Whether the delocalised electrons conduct electricity depends on whether they are mobile, that is, if they can change their momentum. In insulators, delocalised electrons are plentiful too, but no new energy state is accessible to them, so they remain in their usual state, with no global "movement" (a tricky notion in quantum mechanics) nor current.
"Band theory" is the proper tool to understand conductivity.
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Conductive mercury has 1007kJ/mol ionization energy, while semiconducting selenium has 940kJ/mol and insulating sulphur 1000kJ/mol.