Can a third-country manufacturer place CRA products on the Union market without any EU-based operator?
No.
The Commission FAQ explains that a non-EU manufacturer needs an economic operator established in the Union to perform the Article 4 tasks under Regulation (EU) 2019/1020. Depending on the setup, that can be an importer, an authorised representative or, where no other such operator exists, a fulfilment service provider.
section 4.6.1