Studies indicate that the normal range of parenting styles have little impact. But the normal range is grossly sub-optimal. So, this research says nothing about the impact of optimal parenting.

Scientific research on parenting has provided superior evidence-based methods that have not been widely adopted by parents due to poor technology transfer in this area. In fact., it's normal for parents chronically employ methods that have been known for decades to be counterproductive.

Certain behaviors are called "behavior traps". Once they are learned it's hard to unlearn them. The behaviors are intrinsically reinforcing and there is a behavioral barrier to unlearning them.

Not eating veggies seems to be a behavior trap. It's not uncommon for picky eating to start around age 2 become a lifetime habit. Parents tend to use counter productive methods in an attempt to address picky eating. Evidence-based methods are available but not widely used.

I conjecture that optimal evidence-based parenting methods have a huge impact on the outcome of adult unhealthy eating behaviors.