Of course, you'd naturally think the sooner the better! But if you've ever been involved in building a website, you'll know that once you set out to create your own site, you start thinking about making it something you won't be embarrassed to show the customers who will see it after launch, as well as your competitors.
That's a great mindset in itself, of course, but it tends to make you spend 'more time than necessary'. That's why I recommend refining the site after it goes live.
From the client's point of view, the image is publishing at about 80% completion. Specifically, it's the stage where the overall structure and all the pages contain accurate information, but the fine-tuning of the design, images and wording still remains.

And when I actually make that suggestion, company presidents and the staff in charge invariably say, 'Why? I'd like to raise the quality further before publishing.' Of course I understand why they feel that way, and I know it's an unreasonable request for large corporations that require many levels of approval.
That said, for small and medium-sized businesses that need results early on, there is almost no benefit in waiting for the remaining 20% of fine-tuning before publishing.

Why should you publish your website before it's finished?

A website won't rank highly in search results right after launch

If you've ever built a website, you'll know that it almost never ranks highly for the phrases your prospective customers are likely to search right after launch.
In some cases, it may never rank highly at all.

As long as your website provides information that's useful to prospective customers, it will definitely start showing up in search results, unless you're competing for nationwide-scale keywords. But it won't reach the top of the search results within a few days or one or two weeks of launching—no matter how good the content and the site are.

Naturally, if a website has thin content or lacks the basic measures needed to appear in search results, it will never rank highly no matter how much time passes.
But even a website boasting content second to none may find its rankings slow to rise for several weeks, or in some cases several months.
Alternatively, it may rank highly right after launch, only for the ranking to drop after a while and then take some time to recover.

Could the domain aging filter be at play?

Google doesn't acknowledge its existence, but it's a phenomenon anyone who has handled a number of web production projects has experienced at least once.
The domain aging filter is a phenomenon where, after newly acquiring a domain and publishing a website, the site has difficulty appearing in search results for a certain period.
When it occurs, search rankings are slow to rise for anywhere from two weeks to about three months.
There's also the opposite phenomenon, known as the 'Google honeymoon': the site ranks highly right after launch, but then drops sharply soon after.

Neither phenomenon necessarily occurs every time, and the duration varies.
As for how to avoid or resolve the domain aging filter—there's really no definitive method.
If there's nothing particularly wrong with your website, it will start appearing in search results after a certain period.
And if your site ranked highly and then dropped, it will return to its original ranking after a certain period.

That said, when a site doesn't show up in search results, or barely does, clients get frustrated—and honestly, we get anxious too.
But panicking and thinking 'Oh no! Is it a penalty?' is a mistake. It may actually delay the resolution.
To repeat: it isn't a penalty, so your site will properly start appearing in search results.

Think of the first few weeks to few months after launch as a 'trial period'

For these reasons, it's hard to expect visitors from search right after publishing a website.
This is precisely why it's better to publish early.
No matter how wonderful you think the website you've built is, there's a time lag before unknown prospective customers start visiting.
So if you publish when it's nearly there and refine it further while waiting for the rankings to climb, the time until you see results will be shorter.

It's said that the domain aging filter may be an evaluation period during which Google asks, 'Is this website trustworthy? Is there anything fraudulent about it?'
So it's important that we, too, view that period as a trial period for the website—completing the site after a post-launch testing period.

Incidentally, here at Merci we provide free revisions for three months even after launch and final acceptance. The reason, as written above, is that we see our work as not ending at launch, but as including everything up until the site is reflected in search results.