So what evidence is there for it being safe or not?
It appears to me that there is evidence that repeatedly heading the ball causes brain damage.
https://www.nbcnews.com/health/health-news/heading-soccer-ball-harms-women-more-men-study-finds-n896236
and from the JAMA study quoted in the NBC article:
"Conclusions and Relevance: Although heading is the most common activity associated with concussions, the most frequent mechanism was athlete-athlete contact. Such information is needed to drive evidence-based, targeted prevention efforts to effectively reduce soccer-related concussions. Although banning heading from youth soccer would likely prevent some concussions, reducing athlete-athlete contact across all phases of play would likely be a more effective way to prevent concussions as well as other injuries."
https://jamanetwork.com/journals/jamapediatrics/fullarticle/2375128
So, basically, yes heading can cause concussions and damage brain tissue, but it is more likely your kid will get injured from running into another kid on the field.
Best way to keep them from getting hurt playing is to keep them from playing.