While working on this, I realized I could can get most distribution I would like to have as a player. Which makes your method pretty close to a point-buy with a random budget. You can read further to see a comparison with 4d6.
Comparison with Point Buy
Two aspects bring this method closer to a Point Buy. First, I can choose what the shape of the array will look like. Second, I can switch dices around to get better bonus. (For example, turning a 15 to a 14 in order to raise a 7 into a 8. Raising one of the lower bonus at no cost to the higher bonuses)
The system recommended in the official rules is scaled and restricted to stats in the 8-15 range. I have never used it so I can't speak precisely about it. (I am used to non-scaled and restricted to 8-15)
Compared to the official Point Buy, your system allows for higher stats, allow for lower stats (due to lack of limits) and makes it easier to get higher numbers (due to lack of scaling). With the caveat that low stats are actually hard to get because you need low dice for them.
I can compare with the example given in the rules here and try to replicate those arrays in Anydice.
The first example array is a 3-high/3-low that makes [15-15-15-8-8-8]. By adjusting the distribution like this, I get a similar and possibly better array. The low are similar, with a 50% chance of being 8 or better, and probably a 6-7 otherwise. While one of the high are have the same probability of being a 16 or higher.
On the other hand, the average array given is [13-13-13-12-12-12] and my best approximation is closer to a mix of 13,12 and 11 (Here is the AnyDice).
So, compared to standard Point Buy, the method is worse if you try to have an average array. But better as you try to have extreme stats. On top of allowing stats above 15 which are forbidden with point-buy. I would describe it as "not balanced with point-buy, but similar in power" : the important stats are expected to be a few modifier highers and the dumpstats to be significantly lower, but not that out of the norm.
Comparison with dice-based methods
You do not give any ways that you think your players will use your method, so I'll use the way I would do it.
First I try to get my 3 main stats as high as I can. On most characters, this would be my attack stat, then Dex or Con depending on what kind of character I'm playing. Then the rest is divided as evenly to get as few glaring weakness since I do not expect to actually use those stats, I just don't want a glaring -3 on a skill or a save if I can avoid it. Lastly, my race will probably give me a bonus to my first and either my second or third highest stats.
The 20d6 method
Here is what I get with this strategy Anydice. The lines are the chances of getting at least X.
Notice how the two main stats are very likely to be at least 17 and 15 before taking into account races. With the third stat around 12-13.
Also notice how the last three stats tend to be close to one another. Similar to the standard array and 4d6 methods.
On a personnal note, I would never take an ASI over a feat with those numbers.
Comparison with 4d6
For reference, here is the probability curve for 4d6-drop-lowest and 3d6 (Taken from an article on Anydice). I will be basing the following on this curves as well as my personal experience with rolling a complete array with 4d6.
The main stat can be safely expected to be a 15-16 (+3 or better with races). With the bulk of the stats falling between 12 and 14 (+2 for important stats, +1 for most of them). And the lowest being -1 or +0.
Compared to this, your method gives:
- a guaranteed +1 or +2 for the main stat when compared with 4d6;
- a guaranteed +1 for the two secondary stats
- the lowest stats are -2 compared with expected results form 4d6.
I don't think this will be that gamebreaking. But I would feel like my character is a full power from Lvl1. As mentionned, I wouldn't take more than 1 ASI until my build is complete from a feat point of view.
While the 4d6 bell-curve ensures most character have a bonus around +1 in all stats and a main stat around +3 for their specialty, you method gives higher highs and lower lows. More precisely, the highs appear to be slightly better than 4d6 (by +1 usually) while the lows are significantly worse ( by probably -2).
All in all, this method appears less gamebreaking than it does at first sight. It actually looks fun for a high-power game. You will absolutely need feats to make those Lvl4-8-12 interesting. Especially for a fighter.