Väki also means (or at least used to) 'power' or 'strong' (väkivahva, väkijuoma...) and that's also where the etymology for 'väkivalta' comes from. So it's got nothing to do with people.
Valta also sounds like a Germanic loan just a second... yep. Same root as German "Gewalt", violence, "walten", to rule, preside, "verwalten", to administer, also English wield.
What's it with Finnish. One third borrowed from Estonian, another third from the Swedes, the rest from the Sami.