We have a party of five. Currently, three of them have already chosen (independently) and their choices are all different, so there's no problem of one row clearly being 'the best'.
I recently saw a variant of this method on a mythweavers chargen post. They rolled three sets and kept the newest set, not having an option to choose earlier sets.
In the current campaign I'm running, players used a variant of the "Focus and Foible" system that I believe was published in "The Way of the Wicked" for Pathfinder.
The original was pick a stat to make 18, pick another to make 8. Everything else is rolled as 7+1d10. I tweaked it so the Focus was 15+1d4 instead of 18, and the foible was 6+1d6.
I do a roll-around. The player to my left rolls 4d6 (drop lowest) and records the entry on a 6x6 grid. Then the next player rolls. Then the next, until each of the 36 squares is filled in.
Then, once the grid is full, each player chooses a column, or row, or diagonal array of 6 numbers. They may not select the same array, so once it's claimed its gone.
With these numbers, you put your stats in order, you may then swap two numbers.
Anyway. It sounds complicated, but we roll together as a table, then individualize the results.
I love anydice. I looked at that and you avg 12/13s for four stats and 17/18 for your focus and 9/10 for your foible.