I'm wondering whether or not this design will be stable enough.
The design without any front rail should be fine.
Most of the forces exerted during use are transferred directly down to the floor through the legs, and there's more than enough meat in all the rest of the joinery to take care of most concerns about racking.
As highlighted (very much!) in a recent Answer, thick hardwood is very stiff, so there's no absolute need for the front stretcher1 as support for the top, although of course that isn't their only function.
He's a little resistant to this, because he wants to be able to sit in his chair cross-legged, and his current desk (an entirely different design) doesn't have a top rail there.
The customer is always right. Well they aren't but you get what I mean ^_^
but also as an additional area to attach the top - I'll be using Z-clips to allow for wood movement.
Worth highlighting that these clips, like figure-8 fasteners, should primarily or exclusively be used on the sides of a table with a solid-wood top because they minimally allow for movement when oriented in line with seasonal movement as they are front or back, i.e. on the rails and not the stiles2.
From the discussion in Comments:
Adding corbels back into the front is an option that will add support (not needed) but potentially a lot of rigidity to the overall piece. Although the corbels would look good IMO, they are a feature of the original after all, you can add rigidity without any compromise to the clean, open front of the middle design by directly screwing the front legs to the top.
Fixing one edge rigidly to whatever understructure is present isn't normally done in competent work, but is occasionally — where the goal is to have expansion go in one direction only3, rather than inwards and outwards from the centre as is usual.
1 Or either stretcher in fact, as can be seen fairly well from a lot of MCM and later stuff which often eliminated them entirely, and sometimes the rails as well!
2 This is one reason why both types, although neat and unobtrusive, would not be my choice of tabletop fasteners. These are fundamentally a better design as they are agnostic as to orientation, as well as being far easier and faster to install. Tend to be cheaper too (when you can find them).
3 In demilune tables for example, the flat back edge you want to stay put in relation to the wall, so you fix that in place and the front edge then moves back and forth as much as needed.