Ok,
I've just had a look at a lineup of cartridges from 7mm through to 500 NE.

If the criteria includes (no wad) case capacity and bullet availability, I can really only come up with one "gap" and that is the .338 cal which you have mentioned earlier in this thread.
The .333 Jeffery is there but with a slightly smaller bullet dia of .333.

So you could create a .338 maybe by necking down the .375 Flanged Magnum and have a case capacity which wouldn't require wadding, and you would end up with what could be called the .375-.338 NE Flanged Magnum.
John you could do this and call the cartridge the 338 NitroXpress!

If you look at both NE cartridges and Metric flanged cartridges, there is a very good range available which have case capacity well suited to modern powders.

Regarding a .416 with less recoil than the 500-416, there is already the .400 Jeffery which has good case capacity and nowadays has good availability of components including bullets.
You could of course neck it up to take .416 dia bullets and therefore create the 416 Jeffery or the 450-416 NE or the 400-416 or whatever.

If however, a person wants to use relatively fast powders in the existing NE cartridge line up, things may be different.
Trying to use the faster powders, and reduce case capacity to eliminate the need for wadding, and at the same time keep pressures down for use in double rifles, is maybe asking too much IMO.