No, if anything there will be more because some of the funds freed up can be spent on healthcare delivery.

But you will lose endless billing departments at general/specialist practices, hospitals etc, because actual consumers would not engage. Hospital administrations will be driven by the government setting reimbursement rates for services that are transparent, so the argy-bargy between someone paying cash, someone getting insurance etc will disappear (mostly).

I'm in Australia, and the billing departments, even at private hospitals are minimal. Ditto at general practices. Reimbursements for GP/specialists that are not "bulk billed" (ie paid at the government rate) are automated into a nominated bank account and are usually instant.

Medicare for All in the US would move the 2/3rds that are not covered by Medicare, VA, etc into the same systems, with the reduction in overhead from 10-20% to the 2-5% of the current government run health systems.

Allowing Medicare to negotiate both drug prices (as a large purchaser) and health services will also dramatically reduce costs, because the government has the purchasing power to drive hard bargains.