“there is a problem with a single company or organization controlling a nontrivial segment of the voting machines used in the US”

Even if you have many companies providing voting machines, it does not deal with the problem of distrust in national elections. Many elections that have recently occurred come down to the votes of a particular small area of the country or district. These districts are likely to be dominated by one voting machine or type of vote machine purchased by the election board at that district. So, effectively you still have a single or a few voting machines determining elections.