My understanding is that when a form  needs to be "administrator approved", it's
because it contains code. (I have found that when I first upload a form,
InfoPath often tells me that the form could not be activated for browser
completion, even though there is no error reported in the Design Checker. If I
just repeat the upload process, the problem goes away.)

If the form has code and really needs to be admin approved, when you upload it
to a library  it's just acting as a storage location from where the administrator
can download it. The "administrator" needs to go into central  admin (where you
went) and select the option to upload the form into SharePoint from there. What
that does is make the form into a site-collection-level feature.

From central admin, or from the site  collection features page in the site
collection itself, you can then activate  the feature which makes the form
available in the site collection. Once the feature is activated, there is a
content type in the site collection which you can associate to a document/forms
library (in any site in the site collection) to use the form.

I only found this stuff out in the last week or so! I have been building a
solution to the issue that rich text in InfoPath forms  gets stripped to plain
text by SharePoint when promoted into columns. You can work around it with a
simple bit of code in the form, but it does mean the form has to be "admin
approved".