Who handles publishing the website or system?
Publishing is part of the project. You don’t need to hire anyone else or deal with technical settings.
Before launch
The website or system is available at a staging address so you can review everything. It only goes to the final address after your approval.
At launch
I set up hosting, point the domain, enable HTTPS, create the necessary redirects and, when there is an old site, make sure old URLs lead to the new pages so Google rankings are not lost.
After launch
I set up Google Search Console and Google Analytics, submit the sitemap and follow the first days closely to fix any detail quickly.