It's not entirely clear how the process will work. The FAA hasn't updated AC 23-17C in response to the rule changes. I also haven't seen any discussion of changes to Production Certificates.
That said, Part 23 is still "AIRWORTHINESS STANDARDS: NORMAL CATEGORY AIRPLANES"
and there hasn't been any change to Part 21, so it appears that there still will be Type Certificates, and per 21.20 you stll need to show compliance to the relevant standards, which for normal Category is Part 23.
The big change to Part 23 is that it is more performance based instead of design based (what it must do, instead of how to do it). So the actual means of showing compliance will need to change. I don't see DERs and DARs going away with respect to Part 23. I do expect that what they review and how they do it will likely change.
To your point, LSA receive a Special Airworthiness Certificate (based on ASTM Concensus Standards), as opposed to a Type Certificate defined in Part 21.21.