Has anyone taken into account on how obscurity would play into all this?
Because the Underdark is... well... dark. And obscurity is a serious penalty factor in there, which acts as an advantage for the enemy and a disadvantage for the party. It's basically treated as a hostile environment because of this.
So by having night-time, it would primarily boil down to obscurity battles just like the Underdark does now, which seriously limits certain races/classes without being prepared for it.
Yes.
First of all, it's not necessarily true that darkness is "an advantage for the enemy and a disadvantage for the party." Races that have darkvision would do better at night, which includes many (most) player races. If we ever get a non-tadpole Drow or Vampire (e.g., future companion and/or mercenary) then a night/day system would allow them to excel at night but be hampered during the day. Additionally, players have lots of options for light through torches and spells, some of which can potentially even harm/disorient creatures that spend their entire time in the darkness.
Darkness would also make stealth missions much more thematic for the player. Per 5e RAW, even creatures with darkvision still have Disadvantage on perception checks while in darkness, so many players would actually
want the darkness of night for certain sections of the game to make it easier to sneak by enemies. Mechanically, darkness would provide additional options for players, adding to their tookit. This is in addition to all the immersion arguments.