The thing is that a follower is either useful or not. If he's useful, and doesn't cost character resources to get, then he's a free power up or a DMPC. Both of those are Bad Things.
I agree, but I think there is still a way to include followers in the game.
First, think of hirelings. They sell services for pay. (let's ignore pay for the moment and just say coin for easy understanding). When I hire men-at-arms the expect pay to work for me. If I tell them we (e.g. 5 PCs) hired them, then they follow all of us. How much they are worth for their services depends on how much we have to pay to get 'em.
Henchmen are another type of follower who has also signed on, but is usually attached to only one PC. Think of it like a magician's apprentice. Your wizard PC has opted to give (pay) a good deal in order to have that NPC work with you. Part of that is probably food, lodging, a share of the treasure (and not pay), and training on how to be a wizard.
Leadership gets into what rules are in effect relating to the NPCs following you. This can mean morale, loyalty, and other rules. Do something that breaks a henchmen's loyalty to you (maybe a save is involved) and they might leave. Don't live up to your end of the deal or pay the men-at-arms and they will leave to - or turn on you. It depends on alignment.
Followers in AD&D were a class specific benefit. If you picked Ranger, then you received a specific number of henchmen-like creatures (even humanoids) who followed you. I don't believe these were apprentices. They aren't in it for the training, they are adherents to your awesomeness. Classes like this were limited in terms of henchmen they could have beyond CHA too, so it wasn't a best of all worlds situation. And if they left or died, that was it. You don't get any more. The balance though was built into the class itself.
Associates were basically NPCs who have no deal with you, but you've convinced to travel along with you for however long. This could even be the guy in a tavern who's opted to sit and talk for awhile. It doesn't mean they are a DMPC, but they could travel and work with you as long as their current goals were in line with your own. When they stop being so, they go their own way.
I don't know how 5e will handle NPCs and what they will mean to PCs, but I expect a lot of options will be offered. No single rules beyond core sounds like the name of the game. Will Followers, Cohorts, and Leadership be core? I couldn't say.