Nope, I'm not. Or reather, I am working them differently because the triggering conditions differ. I'm using the right calculations for each.

Fighting Spirit kicks off before the attacks. You need to look at the conversion rate for all attacks. Precision kicks off after you see the roll, you only need to work out the conversion rate for misses. Now, if you have no idea what you need to hit, precision becomes more of a crap shoot, but with multiple people and multiple attacks you'll quickly narrow in on the right AC.

The trigger doesn't matter to the math. If you're using precision to correct a miss, then you're in the same condition set as using advantage to prevent a miss -- rolling dice. Once you do that, you have to make the comparison the same, and that comparison is not the adjusted overall hit rate for advantage to just the miss conversion rate of precision.

Let's check the following -- a single round where two attacks are made. Let's assume that the hit chance is 65%.

The advantage case will have a total success rate of 65% + 22.75% or 87.25% chance that each attack is a hit. Yay!

The precision case is slightly more complicated, but what we want to get to, for comparison sake, is the same total success rate. To start with, we take the base 65% hit rate. To add to that, we need to figure out what the conversion rate of a miss to hit is using precision. You've done some of this work. The rest is figuring out what chances actually exist. So, as I said above, we miss on a 1-7. We can toss out 1's -- they always miss. That leaves 2-7. There's a 5% chance for each to have occurred on a roll, but different odds of successful conversion for each, so we can't take averages (like the 51.6% you calculated above) and spread it out -- the distribution is not even. That means the following:

7 -- 100% converts, 5% of the time, so overall 5% conversion rate.

6 -- 87.5% converts, 5% of the time, so overall 4.375% conversion rate.

5 -- 75% converts, 5% of the time, so overall 3.75% conversion rate.

4 -- 62.5% converts, 5% of the time, so overall 3.125% conversion rate.

3 -- 50% converts, 5% of the time, so overall 2.5% conversion rate.

2 -- 37.5% converts, 5% of the time, so overall 1.875% conversion rate.

This totals to a combined 20.625% conversion rate of misses into success. Since this already takes into account the die roll probabilities like the base hit chance, it can be added to get the total success rate. That rate is 65% + 20.625% or 85.625% total success rate. It's lower than advantage, even when precision is chosen to be used only when a miss occurs.

This means, over two attack, the odds of no successful attack, 1 successful attack, and two successful attacks are:

For advantage: all misses -- 1.6% 1 hit -- 22.2% 2 hits -- 76.1%

For precision: all misses -- 2.1% 1 hit -- 24.6% 2 hits -- 73.3%

So, on the narrow point that advantage is superior to precision mathematically for total success rate, yes, you have a conception issue and I stand by both my math and point.

On the matter of precision being efficient, or enabling better action economy, sure, have fun, good arguments, don't disagree. I engaged on the point that the math favors precision -- it does not. End of point.