I think it in fact does make a difference in penetration. If your arrow hasn't straightened out at whatever distance your target is it will be far less effective. You may hit the target at point of aim each and every time but if your arrow hits at an angel your penetration will be seriously diminished.
Keep in mind that this issue can be caused by different factors,
1. Arrow spine matched to the point weight, arrow length and bow weight.
2. Broadhead type.
3. Arrow quality (consistency) and straightness.
4. Your release - torquing, smooth release, follow through.
5. Good anchor point each and every time, be consistent.
In reality .006 vs .003 you may never see a difference in arrow flight. But in my opinion, it lessons one area of concern. For broadheads I go with .003 for my recurves and .001 for my compounds. Could I get by with .006 in my recurves ya probably and .003 with my compounds with mechanicals yup. I won't chance it with fixed heads.
I still think it would be worth your time to get 1/2 dozen of each and test them. Worst case you will have 6 great shooting arrows and best case 12 arrows that shoot the same and your questioned answered with hands on experience.