[Discovery] is certainly not Trek Prime (and not just in style, but also for plot-related reasons) […] nor is is quite Kelvin either
I just watched DSC episode 4, and it is firmly not Prime as far as the tech goes. Apparently, the unique thing about the ship itself is that, in addition to a traditional warp drive, it has a working drive engine that can transport the ship to anywhere in the universe in the literal blink of an eye so long as they have the ability to store and input navigational vectors correctly.
In Trek Prime, the Traveller warped the Enterprise D to the Kalium Triangulum Galaxy—not to mention also what they assumed at the time was the actual edge of the universe—and being able to do so was a downright inconceivability to everyone involved except the Traveller himself (and maybe Wesley). So the fact that Discovery, which existed centuries earlier, has a working drive engine that can outstrip even Voyager or the Enterprise E—that means the show itself can not take place in the Trek Prime Universe.
If it were, Starfleet would have figured out how to implement such a drive in more ships by the 24th century, especially given that the technology they use in DSC was not (reported as of yet to be) stolen from the future so no timeline meddling would have been of concern.