The thicker the material, the more pronounced the cut profile “sawtooth” pattern will be.
This project is a lot like any other layered box with multiple parts per layer, and has the same caveats. Take a more extreme example like my space invaders box:
The “pixels” aren’t squares because of the cut profile, and you can see it if you look closely.
If you zoom in…
I flip-mated these layers to eliminate the gap but the misalignment is all down to the inconsistent pixel shape.
Same thing with the spiral staircase box:
Note the stairs are not 90 degrees, but are slightly sharper, about 88/89 degrees:
Again, no gap due to flip mating and good kerf adjustment but the shape of the cut profile is pretty apparent even at 1/8” layers.
In terms of how it applies here, it makes butting stacked cut pieces against a straight cut edge have a noticeable gap on each layer. From the OP:
This gap is almost impossible to avoid even with registration pins and is exacerbated by having thicker layers.
I even experimented with alternating top and bottom per layer to get a more of a 1/4” “zig zag” pattern instead of a sawtooth pattern. It is a different and maybe better look but it’s still noticeable.
The only way I ever managed to get rid of the gap involved leaning into the cut profile and making each layer a bit longer to offset the cut profile, then make my solid wood panels that butted up against it have an edge profile of about 89 degrees. That makes the gap much smaller but it’s much more design work to scale each corner layer piece, and takes some trial and error to get the profile angle right.
If you don’t mind the gap, thicker layers will definitely speed things up, but that’s the tradeoff.