The thing about removing dump stats is...
Why do you want to do it?
There's several problems that can result so it's important to make sure it's worth the trouble.
One pcs generally need to dump something. It's more important to reaise your best score than it is to raise your lowest - hence dump stats. You don't gain much by simply making that necessity more painful. It's not like players have an awful lot of choice about what their best stats are - they're basically dictated by class - everything else is secondary.
You don't get dumb fighters because intelligence doesn't do enough. You get dumb fighters because they have to be strong, fast and tough. Just about very point buy system has the same problem because the pie still needs to be carved up the same way. (And rolling stats can only avoid this if you strictly roll them in order - or someone rolls exceptionally well).
If you make dumping stats so painful that people end up reducing their prime stats to avoid it then you end up with a range of ability scores which are within 1 or 2 points of each other. Given that means 5-10% on a D20 it becomes worth asking what the point of ability scores isin this case? - as they don't particularly serve to differentiate characters.
Especially when using an ability foursome, there is no dump ability. Every one of the four abilities is a must-have.
Here are some pragmatic ways, how the foursome plays out.
Assuming one is a powergamer, the key ability of the class will have the highest score (Strength-Constitution Fighter, Dexterity-Athletics Rogue, Intelligence-Perception Wizard, Charisma-Wisdom Cleric, etcetera).
After that, each choice is a meaningful decision point, a real decision with costs and benefits.
So, a Fighter has a high Strength-Constitution score. That single choice is taken care of. Now, do I want this Fighter to be a light-armor swashbuckling athlete, or an ingenious tactician from a military academy, or a popular war hero who can rally the troops during difficult times? And many other possibilities become valid choices because they are all good choices. For example, a Fighter can choose heavy armor, thus forego Dexterity-Athletics, and focus on either mental ability. The foursome opens new possibilities of meaningful choices.
So far, the foursome works well.
The typical nonplayer human uses one of two average arrays.
• +0 (10),+0 (10),+0 (10),+0 (10)
• +1 (12), +0 (10), +0 (10), −1 (8)
However, the player characters and various superhuman races use one of two exceptional arrays:
• +2, +2, +2, +2
• +3, +2, +1, +0
The standard is:
• +3, +2, +1, +0
Player races have ability minimums, rather than ability improvements. For example, the Elves of the material plane.
• High Elf: +2 Dexterity-Athletics, +1 Intelligence-Perception
• Wood Elf: +2 Dexterity-Athletics, +1 Strength-Constitution
• Drow: +2 Dexterity-Athletics, +1 Charisma-Wisdom
In other words, in order to play a High Elf, the player using the array must put at least a +2 (score 14!) in Dexterity-Athletics, and at least a +1 (score 12) in Intelligence-Perception.
But there is still a floating +3 from the array, and that tends to make any class a viable choice. So there are High Elf Bards with a +3 Charisma-Wisdom, but they are less frequent.
And of course, the Human can be anything.
So, even without the existence of any dump abilities, there are many good choices.