A short list of the things that most often get missed on launch day, and are embarrassing to find out about from a customer.
Launch day attention goes to how the site looks. The things that actually cause problems in the first week are rarely visual, and most of them take a minute each to check.
Does the contact form deliver?
Submit it yourself, from a phone, on mobile data, and confirm the mail arrives in the inbox rather than the spam folder. A form that posts successfully and silently discards the enquiry is a very common failure.
Is the old site redirecting?
Every URL that existed before and has moved needs a 301 to its new address. Without them you lose the search rankings the old pages had earned, and the links other sites pointed at them.
Is it actually indexable?
Staging sites are usually blocked from search engines. Forgetting to remove that block at launch is the single most expensive five-second mistake in this business.
Does it work on a real phone?
Not the responsive preview in the browser. An actual phone, held in one hand, on a connection that is not your office wifi.
Who gets the renewal notice?
Domain and hosting renewals should go to an address the business controls, not to an agency inbox or a former employee. Sites go down over this more often than over anything technical.