I understand you're looking for the Publish button to update your published project.
If you're not seeing the "Publish" button next to the "Share" button, please try the following:
Click the "Share" button
The modal that appears is split into two sections — in the lower "Publish" section, click "Update"
The help article Update a published functional prototype or web app doesn't seem to explicitly mention this at the moment, but it appears this is one way you can update your project, based on what I was able to test on my end. I'll pass this feedback along to our team internally.
There are no limits on how many times you can publish/update a Figma Site. Regarding the publish button disappeared after updating a component, here are a few quick things worth trying:
Refresh the browser/file completely, then try publishing again
Make sure any library/component updates were accepted in the Figma Site file
Confirm you still have edit/publish permissions (on paid plan, and it requires edit access to the file)
If the Publish button is still missing after that, I’d recommend submitting a bug report so the team can investigate the file directly. To do that: Go to the Support Hub / Type “Report a bug” / Click Start a chat
Please also include:
the affected file URL
screenshots of the missing Publish button/state if possible
and grant “Can View” access to support-share@figma.com so the team can take a closer look at it