Well, I'd think that OD would totally dominate because its ^4... That is a pretty big power. But ok I'll see what I can calculate...

Ok, I ran some numbers with google sheets. I calculated tubes with thickness of 1mm, 0.85mm and 0.7mm. I stopped at 0.7 because titanium thinner than becomes fragile like a beer can.

Since I have specific bikes and differences in mind I calculated moi for a 44mm od (downtube), then my raijin's 19mm seat stay and the r330's 15mm seat stay, each with the 3 different tube thicknesses:

Oh. Bummer. I made a nice table as a jpeg but can't share it because I odn't have enough posts. Hmm.

Ok, here is a text version you'll need to suffer through. The big number in the 3rd column is the moi which is linear with stiffness:

OD (mm) Wall thickness (mm) MOI % of 1mm
44, 1, 31239.21195, 0.00%
44, 0.85, 26828.08678, -14.12%
44, 0.7, 22322.10272, -28.54%

OD (mm) Wall thickness (mm) MOI % of 1mm
19, 1, 2297.289628, 0.00%
19, 0.85, 2000.13894, -12.93%
19, 0.7, 1687.120785, -26.56%

OD (mm) Wall thickness (mm) MOI % of 1mm
15, 1, 1083.064067, 0.00%
15, 0.85, 949.1010539, -12.37%
15, 0.7, 805.7595322, -25.60%

Numbers show that you get less than 30% difference between 1mm and 0.7mm wall thickness. A change in tubing OD from 15mm to 19mm is much more significant.

So it looks like indeed OD is totally sufficient for an eyeball of tubing stiffness. Sure you might be able to use crazy thin or crazy thick to change the numbers, but I don't think you can make up a significant difference in tubing OD with wall thickness.

I've got no clue how tubes interact to make a frame feel a certain way, no bias about materials. For frame feel you need to right it, or ask people that do this for a living for advice. Don't listen to me!