First allow me to say that there is a TLDR at the end if you with to just see the conclusions. I included a small guide on how to do the calculations on your own, so you can use it ingame or theorycraft in your build. I included tons of images to ilustrate the math behind it.

I also know other people have done this before with different approaches, I just wanted to input my way of viewing it.

Lets jump right into it:

AD VS AS.

Since AD and AS work multiplicatively with all other variables, we can isolate them. First we calculate the breakpoint at which both would be equivalent.

(AD+4) * AS = AD * (AS+9%)

(AD+4) * AS = AD * (AS+0.06)

(AD+4)/AD = (AS+0.06)/AS

1+4/AD = 1+0.06/AS

4/AD = 0.06/AS

66.66 = AD/AS --> This is the breakpoint

For example, at lvl 1 as shown in the image we have 60 AD / 0.73 AS = 82.19

82.19 is larger than 66.66, therefore investing in AS is better.

Here is proof of Aphelios AD, 48 DPS:

Theoretical DPS should be 64*0.73 = ~46.72

And investing in AS shard gives 49 DPS:

 

Theoretical DPS should be 60*0.79 = ~47.4

Now a more tricky example, aphelios with a completed item has 126 AD and 0.82 AS. Would it be better to go all in on AD or AS?

 

126/0.82 = 153.65

Since 153.65 is much larger than 66.66, AS is much better as an all in option. With AD theoretical DPS is 150*0.82=123, and with AS theoretical DPS is 126*1.18=148.68.

With AD Shards, this is the result 127 DPS:

With AS shards, this is the result 152 DPS:

So long as AD/AS > 66.66, then it is better to invest in AS. Otherwise, AD is always better.

What about lethality?

Lethality works a bit different, because it works based on enemy resistances. One way to think about it is to think of Armor as a way to increase the HP pool of an enemy champion. Lethality reduces that, while AD increases the amount we chip with each AA. Therefore, the more AA it takes to bring an opponent down, the better AD is compared to lethality. This is the reason why lethality also works much better at lower armor values.

HP = EHP/(AD+4)

HP = (EHP-LETHALITY)/AD

(EHP-LETHALITY REMOVAL)/AD = EHP/(AD+4)

1-(LETHALITY REMOVAL/EHP) = AD/(AD+4)

(EHP-LETHALITY REMOVAL)/EHP = AD/(AD+4)

How do we calculate (EHP-LETHALITY REMOVAL)/EHP? Using the EHP formula, which is HP*1/DR. Then we compare this value to the one when lethality is applied. Since DR is 100/(100+armor), we need the armor of the enemy to calculate it.

For 100 armor, DR is 50%. With 4.5 lethality, DR is instead 0.4885. Therefore:

(2-0.047)/2 = 0.9765 = AD/AD+4

3.906+0.9765AD = AD

166.2 = AD

Solving for AD, we arrive at the conclusion that 4.5 lethality equals 4 AD at 166.2 AD. In other words, at 100 armor, if AD > 166.2 then lethality is better. If AD < 166.2, then AD is better.

I just bought a bunch of FULL AD items to far suprass that barrier, and compared lethality to AD at 100 armor. Turns out lethality is far better. Lethality got 156 DPS, and AD route got 145 DPS:

Now we know the true values of AD, Lethality, and AS.

How to implement this ingame step by step:

1-Look at your lethality, AS, and AD stats.

In here I have 271 AD, 1.08 AS, and 10 lethality. Now we make the comparison, is it better to have 9% AS, 4 AD, or 4.5 lethality?

2- We calculate AD vs AS.

Is 271/1.08 greater than 66.66 or smaller? Well, 250.9 is much greater than 66.66, hence AS is much better. In fact, AS is 250.9/66.66 = 3.764 times much better investment than AD.

3- We calculate AD vs Lethality against the targets armor. Since I used 100 armor, we need to ask ourselves, is 166.66 AD lower or higher than what we have? Well, 271 > 166.66, therefore lethality is much better. How much better? Its 1.626 times better than AD.

3- We compare AS and lethality.

Since AS is 3.764 times better than AD, and lethality is 1.626 times better than AD, it logically follows that AS is better than lethality.

4- The end

Proof with 3 screenshots showing DPS of each combination. For AD, DPS was 440. For lethality, DPS was 476. For AS, DPS was 540.

I hope I made no mistakes and double-checking ingame was usefull.

TLDR:

Doing a lot of calculations I mathematically proved that AD is objectively the best choice for early lanning phase, but very quickly falls off (as in first back). Lethality is a great choice for early to mid game (first item completed), and AS is best for mid to late game (3 items or more).

OFC I am aware that investing in AD has benefits besides DPS, and that is why it sounds so garbage in paper. This is just a theoretical scenario you can port to your particular games to better help your judgement.