A quick likely rhetorical fiddly system question for D&D 5th ed.

I was just making a Character for  our  D&D 5th ed game tonight. Jay is running the module from the starter set and I am playing a  (GHASP) fighter... (I generally always play fighters.)

Looking over the fighters Fight styles I noticed "great weapon fighting."
Re-roll damage rolls of 1 or 2 keep the result of the re-roll. Sounds fine, I'm taking it.

Given the systems focus on advantage disadvantage wouldn't make more sense to handle great weapon fighting in a similar way and always roll 2 damage dice then keep the higher, avoiding having to take the time to re-roll?  Mathematically it's different, but I'm not taking the time to figure out the exact differences, I think rolling 2 dice would be aesthetically more pleasing.

Regardless.
Here is the character by the way .
For those about to die we salute you.

My D un D guy.

Name: Orin Holzknecht

Race Human +1 all stats
Class : Fighter
Level : 1
AC: 16
Hit Points: 14

Method 4d6 drop the lowest straight up and down, no re-rolls.
Strength : 14 +2
Dexterity: 15 +2
Constitution: 18 +4
Intelligence: 13 +1
Wisdom: 11 +0
Charisma: 14 +2

Equipment:
Chain mail
Dungoneers Pack.
Pot
Shovel
Pouch
Glaive heavy , reach , 2 handed, 1d10+2
Scimitar: Finesse, light, 1d6+2


Gold: 130
weight 55


Skills
Animal handling
Survival
Artisan tools: wood working

Abilities
Fight style: great weapon fighting (re-roll 1 or 2 on damage)
second wind 1d10+1 haling needs a rest to recharge. (really?)

Background:

Alignment: Neutral Good.
Folk hero
(random rolls)
Defining event: A fey creature gave me a blessing and reveled my secret origin.
Personalty trait: Thinking is for others I prefer action.
Ideal: There’s no good in pretending I'm something I’m not.
Bond: A proud noble once gave me a beating so I will take it out on any bully I find.
Flaw: I have a weakness for the vices of the city particularly women.




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