So let's say you're a Fighter with a Greataxe. This feat is fine at level 1 when you're making one attack on each of your turns. You hit, you roll two d12s and use the highest. Easy.

Now imagine you're level 11. You're wielding a Flame Tongue Greatsword that deals 2d6 slashing plus 2d6 fire, and you're attacking at least three times on your turns. Every time you take the attack action you have to decide which attack to use Savage Attacker on, and when you do you're rolling eight dice in total and tallying up two kinds of damage twice. With the deciding and the rolling and the adding and the comparing, your turns are noticeably slowing the game down, you finish each combat feeling like you've just sat a math test, and all you've got to show for it is an extra one or two points of damage per round.

It's not worth it. Not for any character you're going to play past level 5 at least. Take literally any other origin feat, you'll have way more fun.

If you're really set on playing a character with military experience, and feel like you have to take this feat because it's part of the Soldier background - don't. Just take the Farmer background instead, say your character grew up on a farm and enlisted as soon as they were old enough. You'll get a bunch of extra HP from the Tough feat, and the word "Farmer" appearing on your character sheet won't limit your roleplaying one bit. And if your Dungeon Master says otherwise, tell 'em I said they're being a jerk.

After using this feat on my barbarian, I think its its really fun and while yes its once per turn and doesn't let you reroll for those low damage numbers, what this feat is really good for is for maximizing your crit damage.

For example, lets say you have more than one attack on your turn, That means you have more than one chances of getting a crit, which you can use your savage attacker to make sure that crit does a lot more damage than on average, cause there were a lot of times where my crit damage was lower than my regular hit damage. Even if you don't land a crit, you can always use the feat on the last attack you make, so there's never not a reason to use it.

Additionally, since its once per turn, it works on your opportunity attacks and other reaction attacks, so you can always make sure those attacks deal the highest damage possible

All and all, if you are more of an optimizer than maybe this isn't a good feat, but if you just want to live out the fantasy of having a powerful hitting attack in the most crucial of moments.

This is a surprisingly ineffective feat - running a Monte Carlo simulation on a 7d6 roll, it only gives you an average of about 2.5 damage increase.
Feel free to let me know if I've got something wrong.

public class Prob {
	
	public static void main(String[] args) {
		int diceCount = 7;
		int diceType = 6;
		int iterations = 100000;
		
		long accumulator = 0;
		for (int i = 0; i < iterations; i++) {
			int roll_1 = roll(diceCount, diceType);
			int roll_2 = roll(diceCount, diceType);
			
			accumulator += Math.max(roll_1,  roll_2);
		}
		
		System.out.println (" Average = " + ((double)accumulator / (double)iterations) );
		
	}
	
	static int roll(int count, int type) {
		int accumulator = 0;
		
		for (int i = 0; i < count; i++) {
			accumulator += (int)(Math.random()*type+1);
		}
		
		return accumulator;
	}
}

Unfortunately you can't consistently use it on the last attack you make, since you might miss with that attack, in which case you've 'saved it' for nothing.

Good point about using it for opportunity attacks though. 'Once per turn' means you can use it on your turn and someone else's turn in the same round, so that does improve its value a little.

Why have they removed the ability to use this on Unarmed Strikes? Or are fists considered weapons now?. It used to be that unarmed strikes are weapon attacks but are not attacks with weapons. Has this changed?

Does this work for ranged attacks (like with a bow) as well as it does not specify melee?