@Hercanic
While I love Dark Souls, I wouldn't recommend trying to shoe-horn it's combat system into things without serious consideration. The reason Dark Souls even works in the first place is due to real-time combat allowing the player to input commands at any time with no menu. This lets the game really test the player by adding timing as a skill to test. In a turn-based system this simply isn't possible since the player has infinite time to thinking things through and since the player's physical hand-eye-coordination and reflexes are not being tested any skills or attacks are nearly guaranteed so you cannot have the stark "do or die" style.
Overall, I would like to see RPGs evolve to let turn-based combat offer more challenges than just the "Did you remember to use Ice instead of Fire?" paradigm that many games rely on.
That said, I think people are way to hung up on Earth/Wind/Fire/Water when we talk about elements.
Literally anything can be an element. All an element denotes is that some attacks have properties which are different from other attacks.
Fire Emblem uses a Weapon Triangle. Swords have an advantage against Axes but are weak to Spears, Axes are strong against Spears but weak to Swords, and Spears are strong against Swords but weak to Axes. A bad match-up isn't impossible, it just means you are more likely to be hit and will take extra damage if you are. This encourages the player to carefully consider which units to attack with and be mindful of enemy positioning. There is nothing worse than sending your Sword guys to mop up some Axe users only to realize too late that this places them in attack range of enemy Spear users. Many units can also equip more than one weapon type after Promotion to make them more versatile.
Fate/Grand Order uses a similar method but ties the advantage directly to unit class. A unit attacking a class that it is strong against will deal "Effective" damage (gain bonus damage) and will "Resist" (take less damage) attacks by those units and will likewise deal less damage to and take more damage from classes that it is weak to. Of the Knight class Servants, Sabers are strong against Lancers but weak to Archers, Lancers are strong against Archers but weak to Sabers, and Archers are weak to Lancers and strong against Sabers. Of the auxiliary classes, Riders are strong against Casters but weak to Assassins, Casters are strong against Assassins but weak to Riders, and Assassins are strong against Riders but weak to Casters. The Berserker class is strong against everything but is also weak to everything and the very rare Ruler class deal Effective damage to nothing but Resists all of the other classes (barring special event classes not available on NA servers yet).
On top of that, all Servants have 3 types of attacks - Quick, Arts, and Buster. Quick attacks generate more Critical Stars (which get distributed every turn to determine attack Crit Chance), Arts attacks fill up the Servant's Noble Phantasm gauge (basically their Ultimate), and Buster attacks just deal lots of damage. This gives each attack type a niche - Quick attacks are good for setting up high critical hit chances for next turn, Arts attacks are good for setting up battle-changing skills a few turns down the line, and Buster attacks are good for killing stuff right now. But wait, there's more! The first attack used in a round adds its effect to all other cards after it while cards later in the chain have increased effect. If you open with a Quick attack, then you can use 2 Arts attacks and still get some extra Critical Stars...or you can use the Arts attacks first and save the Quick for last so you have 3 attacks charging the NP gauge while still getting some extra stars. Using 3 of the same attack in a row - even if the attacks come from different Servants also creates a Quick/Arts/Buster Chain and grants an enhanced effect for the round, while using 3 attacks of any type from the same Servants creates a Brave Chain which grants them an extra attack (this is the only way to get 4 attacks in a round). And yes, they stack.
This typing gives the player a lot of choices. Servant class "elements" makes the player consider which Servants to bring along on a given mission. If the forecast says expects Sabers, you might not want to go with a 3 Lancer team. Giving the different attacks their own unique traits also makes the player consider which of the attacks to use. You can definitely go with a 3 Buster Brave Chain and let one Servant completely annihilate a dangerous enemy in one round...but that means you have no Critical Stars next round and haven't charged your NP gauge at all.
Just by giving classes and skills special properties you can give the player choices. And if your system is good, each of those choices will have advantages and disadvantages that make them all worthwhile. By having all of the options be desirable it lets the player get a lot more engaged since they have real choices to make instead of "fake choices" where only one option is actually good in a given situation. And that engagement is why elements are awesome when used well.