Just because Prince Harry and his children don't seem to spend time with Prince George, Princess Charlotte, and Prince Louis, doesn't necessarily mean he doesn't want his kids to be like them. In fact, according to Hello!, it seems Harry is raising his kids similarly to how his brother and his wife, Princess Catherine, are bringing up their brood, despite their family feud. For one, William and Catherine have made it a point to shield their children from the spotlight by moving them out of London to Windsor. Harry did that on a grander scale, by deciding to raise his children in California. It's also worth noting that both of their new homes are surrounded by nature so the kids can play outside.
It seems that Prince Harry also took a cue from William when naming his daughter, Lilibet. That's because William's daughter's full name is Charlotte Elizabeth Diana, after Queen Elizabeth II and Princess Diana. As for Lilibet, Harry and Meghan Markle explained in a statement, "Lili is named after her great-grandmother, Her Majesty The Queen, whose family nickname is Lilibet. Her middle name, Diana, was chosen to honor her beloved late grandmother" (via Archwell).
Despite the similarities, Prince Archie and Princess Lilibet's upbringing is still different than their cousins, since their father walked away from royal life. Regardless, it's safe to say Harry will always love William's children, and won't let distance or tensions change that.