The key issue here is that while most US-based airlines assign seats at the time of booking (so you can pick seats and move them around at will), BA charges an extra fee for this.

So either travel a US-based airline, or shell out the money to prebook your seats.

Last year, for long and complicated reasons, our family was traveling together, but my wife and I were on a separate booking from the kids (aged 8 and 5 at the time). On our way home, we reached Heathrow to discover that BA had upgraded us to Premium Economy but left the kids in steerage. Thankfully they were able to seat us together in the end... but now we're spoiled by the Premium Economy experience