I sympathise with your elder Ds having being in your younger Ds position myself. We had a shared playstation, years later we had a shared playstation 2, but when the playstation 3 came out my parents couldn't afford one for us to share. My sister got one about 6 months after launch with her own birthday/christmas saved money. It was in the dining room as her bedroom was tiny and no room for it in there. She generously let me have a go if she wasn't using it and i asked if i could have a game. (i was 16/17 and she was 18/19) but the big problem was there wasn't much storage space for saved games and with 3 of us having our own save files (she also let my dad play on it) the save space got used up. When i was 19 i bought my own TV and ps3 for my bedroom and deleted everything of mine off her ps3 so she was just sharing with my dad, but he played maybe 2 or 3 times a year if that.
Back then though online multiplayer games or games which are basically never ending (like GTA for example) weren't a massive thing so people were happy with an hour every other day or so.
These days games like fifa, GTA, call of duty and so many others can be played for hours and hours and not hit the end of the game, or can play with real life friends and have headsets to talk whilst playing, and it does make you want to play for longer more often. I ended up playing for at least 4 hours a day when i got into an online group who played together daily.
The fact is your eldest son bought this console, so it is at his discretion to allow younger brother to play at all. You already said he dislikes it and wants it as something that is just for him. Younger son sees older bother playing, it looks fun, i want a go etc and it doesnt seem fair to him he's not allowed.
Honestly i think you need to honour DS1's wishes that it is his console, and it is only when he is willing to let DS2 have a go that DS2 can play. I don't agree at all with have him buy half off DS1, that's not fair. Absolutely split TV usage equally, so each can have control of tv for say, 2 hours a day (or whatever limit you decide), and either have it set DS2 get it 3-5 and DS1 5-7, or alternate whose "go" of the TV is first, or let them decide based on the day if they want to watch TV or not. Maybe, if there is room for a second console (could it be stored under bed etc when not in use?) let DS2 save up for his own, then when it is their turn, simply switch which console is plugged in. Whether they share games or DS2 has to buy his own (more age appropriate/different interests) is less of an issue than the fact DS1 should definitely have it respected that he saved for, bought and owns the current console.