Peer review is not supported by publishers. Many editors are unpaid too.
The peer review process is almost entirely coordinated by unpaid associate editors. They make initial manuscript assessments, solicit reviewers, and moderate the review and response process.
"journal staff moving papers through the peer review system" may happen at a small number of prestige journals such as Advanced Materials, but for most Q1 journals it is all volunteer work. That is the business model that makes companies like Elsevier billions.
Also, are publishing monopolies like Elsevier really all that necessary?