Much is due simply to him wanting to. Any ex-leader could pull the spotlight, but everyone else chooses not to for the good of the party.
Any President is automatically the head of the party. A popular former president still has a great deal of influence as far as endorsements. Even a former candidate such as Hillary Clinton or Mitt Romney matters as someone once chosen to lead the party. Most are aware of this and are careful. Obama didn't endorse anyone (for president). Clinton knew she'd have a strong shot at the 2020 nomination, but might be a weak candidate so early on ruled out a run to let in new blood. Likewise, Biden, as the former VP ("it's Obama's Dad!"), waited 3 extra months to enter the 2020 race after no-one had pulled ahead.
Over to Trump, he wasn't a Republican until 2010 and gladly breaks Reagan's 11th commandment "thou shalt not speak ill of a fellow Republican". What does he care about seeing the Republican party be strong for generations to come, and so on? A normal distinguished former president supports rising stars, and gives them room to grow. Why would he do that? After the Tea Party (and the election of Trump) the Republican party is in a bit of a crisis, making it easier for him to keep attention, but otherwise he's not doing anything that any other ex-leader could do, but doesn't.