To me, a chair stretcher doesn't seem like something that ought to require a lot of design work, but I managed to spend 3 days trying to work out something that I think harmonizes with the side pieces.  See what you think.

The leg assemblies will be angled from front to back by about 9-deg on each side, making the base narrower at the back.  I think this helps to remove some visual weight at the back of the base, but hopefully without placing the back legs so close together that the chair would tip easily when leaning to the side.  What I had to work out next was to splay or not to splay.   

Since the leg assemblies of the inspiration chair were perpendicular to the ground (or nearly), I mocked-up that option.  I thought that possibly the 9-deg angled sides coupled with the rake of the front legs might give the illusion of some splay.   Maybe it did a little, but not much.  I so wanted this to work because I thought it would make the compound angle cuts on the stretcher so much simpler.  But, as hard as I tried, I just could not force myself to like this configuration.    The chair shell might as well been sitting on a milk crate.  The vertical sides just weren't right.

I had no problem with the side view, but from that view you only see the nice curves of the base.

I made a set of angled braces to allow me to mock-up a 15-deg splay.   Now the splay of the legs almost lines up with the edge of the backrest of the chair shell.  This was the easy winner for me.   I asked my wife to take a look at it, and she was in complete agreement.  I knew she would - I'm a very agreeable person. 😁

Not too much difference when viewed from the side compared to the non-splayed configuration.

In the pictures above, there are wood scraps in place that elevate the chair shell above the base.   The idea is to have the front and rear stretcher have integrated supports to give the seat a floating look.

Next, I worked on figuring the compound angles for the cuts on the front stretcher.  The miter and bevel angles are driven by three angles - the 15-degree splay, the 9-degree, and the rotation of the stretcher away from vertical (which, to make the face of the front stretcher roughly parallel to the front edge of the front leg, it needed to be rotated back 22-degrees).  I tried to puzzle out the trig myself, because it shouldn't be terribly hard.  But eventually I gave in and searched for an online calculator.  The one in the screenshot below did what I needed once I worked out the signs for the angles and that calculated blade tilt and miter angle needed to be swapped.  There is probably a way to enter the angles (complementary angles), but I didn't spend much time trying to figure that out.   

I cut the angles on the miter saw, which worked pretty well.   I did use a digital angle gauge for the blade tilt since the built-in gauge is so tiny.  Here's I'm testing the fit of a scrap against a leg.  With the scrap tilted back 22-degrees (roughly) and the end flush against the leg, it was level.

With a longer scrap, I cut both ends and fitted it in place.  I love you, little online calculator.

From there, I made a series of different templates, each with some revisions.  I started with a scrap that was wide enough to accommodate supports for the seat on the top edge.  What I found was that to level the seat, the support on the right had to be lowered because of some unevenness on the underside of the shell.

The next template had taller supports, so that the difference in right/left heights will be less noticeable.

The straight edges on the stretcher are just jarring.   I needed to incorporate some curves into the stretcher so it didn't look out of place with the sides.  I started sketching some curves on the last template.  It was at this point that I started measuring the ends of the stretcher, each to receive a pair of mortises for dominos.  I found that for 8mm dominos, I would risk blowing out the bottom of the stretcher (since the dominos will be perpendicular to end surface.   I needed another template to widen the end to give more room for the dominos.  With the lever arm of the splayed legs, these will be the most stressed joints on the chair, so I want to ensure it's a strong joint.

At this point, I made the final template on the computer to ensure it was symmetric, added generous curves on the top and bottom to lighten the look.   When I was satisfied, I printed it out and glued it to some 1/4" MDF.

I was confident enough at this point to go ahead and make the oak stretcher.  I cut the compound angles on a blank at the miter saw, traced the template on the oak, trimmed close to the line at the band saw, and finally attached the template with double stick table and trimmed to shape at the router table.

Checking the look.   Is it just me, but does that opening in the center between the two supports along with the seat shape conjure an image of a stingray or manta ray with its mouth open?  (Thanks, Ron, for the sea creature hypnotic suggestion.  I'm on path I didn't realize.  😆)

One (final?) adjustment I made was to remount the template to the stretcher and shave about 3/16" off the bottom - same curve, just shifted up 3/16".   I was just looking a bit heavy.

Next up is getting the mortises cut in the stretcher and legs for floating tenons.