Why Can't I use an Organization Profile type in a Contribution Page?
You kind of have two questions here: the second being an issue you ran across after addressing the first.

To start with, yes, the main profiles on a contribution form are limited to contribution fields, individual fields, and fields for all contact types.

The presumption is that only individuals have fingers to fill out a form; even if the form is an organization's membership or donation, it's really an individual acting on behalf of the organization.  However, you already solved this: you started using the "on behalf of organization" options.

I know the feature you describe with the cid=0 for filling the donation form as someone else, but that box should only allow you to select individuals, since it's filling the parts of the form that are individual-specific.  Meanwhile, it wouldn't necessarily make sense to fill the organization fields, because an individual may have any number of relationships with organizations.

If someone were to extend CiviCRM to add a second box for selecting the organization that is related to the individual making the contribution, that would be a nice feature.  However, it doesn't do it at the moment.