Unfortunately not the case, you have to do both EMC and RED for a product with radio functionality.
Again in tldr: EMC as generic term is always needed. As a directive name, if RED applies then the RED contains the EMC stuff.
Yes, to add to this: that's why I wrote that the EMC directive is not applicable. You would fail the EMC directive as you have a radio.
Yes. If there is any radio in your product, you must comply with the RED directive.
Even if you use a ready-made radio module that claims to be "compliant". A very common source of misconception.
Yeah. It's grey area if you could skip some of the radio measurement parts of RED, if you can get testing reports from the module manufacturers. But definitely you need to still comply and test for everything else in the RED. Savings from skipping a few radio measurements are not that significant anyway.
For some parts of CE, you only need technical documentation that you are compliant with the standards, and the documentation is only internal.
This is to make access to the market easy, and cheap. So that's why the documentation can be a single page, derived from other companies. Saying like "The radio module is certified, I'm using 9dBm for the TX power, and the PCB is good" in theory is an acceptable documentation for RED. You don't necessarily need measurements, and for a product sold in small quantities by a startup for a non-critical application it might be enough.
If the product is actually proven to be non-compliant, thats a whole other story.