Since there are no guesses for the 3rd team with 5 players 24-and younger with 2000+ minutes, here's the answer: the 2013-14 Utah Jazz. Jazz had great health that year, so all of the "Core Four," plus Trey Burke (and even non-youngster Richard Jefferson) had more than 2000 minutes. So to summarize the three teams:
2023-24 Spurs (Greg Popovich/Wemby led); finished with 22 wins; draft prize: Stephon Castle
2022-23 Rockets (Stephen Silas/Jalen Green led); finished with 22 wins; draft prize: Amen Thompson (probably the youngest of these three teams)
2013-14 Jazz (Ty Corbin/Gordon Hayward led); finished with 25 wins; draft prize: Dante Exum
2023-24 Spurs (Greg Popovich/Wemby led); finished with 22 wins; draft prize: Stephon Castle
2022-23 Rockets (Stephen Silas/Jalen Green led); finished with 22 wins; draft prize: Amen Thompson (probably the youngest of these three teams)
2013-14 Jazz (Ty Corbin/Gordon Hayward led); finished with 25 wins; draft prize: Dante Exum