So I thought using the multisite option in WordPress would be the most common practice for creating a website for a client to review via the web while it's still in the development stages and then rolling it out to the public once it's been approved.
Yep. Tons of people do. Why, I even wrote a handy article all about it:
While it says wpmu, about 90% or more is still relevant & pretty much the same.
Upon approval I can move this site to it's own hosted environment as the primary domain
And that's the sticky point. :) If it;s staying in that install, you could map the domain to the site, where it sits. Easy-peasy.
Extracting it out, with all the settings, is a little more work. Do-able, but a wee bit more careful than you woudl in a single install. That's all.