I don’t know if you’ve experienced this, but when you nail the back board on a bookshelf, it ends up much more rigid than before. It doesn’t take much material to really stiffen up that dimension. In addition to keeping the box from racking, the resistance to tension and compression also make the whole thing resistant to bending. I certainly hope the bottom will add a lot of rigidity and I also hope screwing the top in would add a lot more, even without glue.
I think the only requirement for a lamintated panel to have the desired stiffness is that the layers do not experience a shear failure under load. However you accomplish this is not really relevant. Consider a hollow-core door: It has a solid wood perimeter which is glued to the skins (and prevents shearing) but the interior is a very loose honeycomb of brown paper which only needs to resist compression (keeping the skins apart) and sees almost no shear load whatsoever.
With that in mind, simply screwing down the skins around the perimeter will give you basically the same stiffness as gluing the whole thing, but you won’t get as much ultimate strength (since you will be limited to loads that don’t start tearing the skins through your fasteners).
True, but that is mostly for keeping things in place (honeycomb from moving and the skins from bowing and not to add any significant strength vs. shear loads. Look at the interior of these doors: https://www.youtube.com/watch?v=KIwD2VDpsrY and imaging how much load they would bear without the solid blocks on the ends.
Late to the party but wanted to point out that surfacing an MDF board can cause some problems if you are going for a lot of Z accuracy. The compressed outside layer is more moisture resistant so when you face it you may notice it will start swelling in those spots. The most recommended option seems to be using Zinnser BIN shellac based primer to seal cut edges. I was able to get within 0.4 mm across a full sheet table with shimming the Z carriage and paper shims glued under the spoil board but I could see maybe facing it to get that last bit of accuracy.