FWIW, here is where the research is. (This is in my field).
Screen time before 2 is mildly correlated with delayed language learning. Children OVER 2 may show some benfit from educational TV. However, there is evidence that even PBS shows teach children relational aggression.
Adult programming on in the background disrupts children's play. Though they may not be watching, they will turn and stare and get distracted. Parents also speak less to children when the TV is on.
Screen time in general is correlated with obesity and disturbed sleep (this is especially true when a TV is in the child's bedroom or it's watched right before bed).
I don't know how any of this varies for gifted kids. We have been very restrictive of screen time for our kids and feel pleased with the end result, but they might have been this way anyway. I really appreciate that I am not constantly being nagged to watch TV or play videogames. My kids spend a ton of time outside just playing.
