The unpublished redesign has a special status: it cannot fail, because it cannot do anything. Somewhere in that comfort, weeks pass, and the old site keeps greeting every customer the business earns.
Drafts do not collect evidence.
A draft collects opinions. A live page collects evidence: which services people actually read, which questions still get asked by phone, where visitors stop. The fastest way to find the real problems is to let real customers meet the page.
Perfectionism also misjudges the audience. Customers are not grading the site against the version in your head. They are trying to understand the business, trust it, and take a step. A page that does those three things is ready.
Live is a starting point.
Publishing early only works when changing the site stays easy afterwards. If every fix costs a request to someone else, going live is a deadline; if the owner can change a sentence the moment it reads wrong, going live is a starting point.
That is the real order of operations: publish the honest version, watch what customers do, and improve the page the evidence points at. The site gets better in public, which is the only place it matters.