Yes, I think you’re right! The thinner pieces don’t have enough strength.
I would seriously consider making walnut veneers about 1/16 to 3/32” thick IF you have a well tuned bandsaw. Then glueing those to a lumber core you have made from strips of quartered yellow poplar ( for us yanks that is actually a magnolia, not poplar) or pine.
I recently did this for the top of a small demilune table.
You also don’t need a vacuum bag, you can hammer veneer it to the core with hide glue.
I’d love to hear how it progresses, Ed.
Best of luck