Your stats are basically chosen for you if you need exactly two good stats and can safely dump everything else, which leads to the current observations of homogenous stat arrays. That's the problem. Rogues need Dex and Con, wizards need Int and Con, warlocks need Charisma and Con, etc. A couple of classes, such as paladins and monks, do actually face a meaningful choice between Constitution and another useful stat. It would be nice if more classes worked that way.
As some sort of platonic ideal, if every class benefited equally from every stat, then you actually could build however you wanted without being penalized. If you wanted a fighter that was good at lore or social skills, you wouldn't have to shoot yourself in the foot in order to do so. I'm not sure what sort of benefit could make Intelligence or Charisma as valuable to a fighter as Strength or Dexterity, though.
That's not a fun choice to offer the player. As a designer, you shouldn't ask the player to choose between being efficient but boring or weak but fun. All characters should be fun.
And that's a great reason to remove Constitution from the game. If fighters didn't need Strength (or Dex) and Con in order to do their job, then they'd be able to quickly cap out their prime stat and then move on to Int or Charisma or something. It definitely helps if you're not using feats, of course.
There's a number of issues I see here:
1) All but a few classes NEED a decent con mod. As I'm mentioned before 5/7 of the Critical Role Team have CON as their 2nd Lowest stat and their game is doing alright....By your definition, Toughness should always be taken as it's equivalent to 4 points of CON (+2 Mod). Yes, if you are playing a certain type of game optimally you are going to prioritise CON over other things, and in those sort of game s those 'other things' aren't going to come up as much.
2) Yes, we all have our own biases about how the game should be played, but they are personal to us. You say you can "safely dump everything else", or a "Meaningful choice" over other stats. Then you say a Fighter would have to shoot himself in the foot to pump Intelligence. Therefore, I can infer that you think Dumping social stats is "Safe" and not have CON as your Second stat is "Penalising"....Fine, that's your opinion, but it does show your bias. I would say a Fighter/Barbarian/Rogue with 12, 10, 8 in Mental stats *is* being penalised in every aspect of the game that isn't being punched in the face. It's only a safe choice is your only measure of the game is dying through HP degradation. Again, in certain types of games that's totally fine (and with certain GMs, a very valid concern). In other games, you are damaged because of it.
(I've now got this hilarious situation where your party Wizard has been kidnapped, but no other character has a high enough Intelligence to read the ransom note).
3) This isn't about being efficient but boring or weak but fun. That's a false dichotomy, because the GM can tailor encounters and the game as they want, and players themselves get joy out of different things. Because in D&D the difference between your 'Efficient' and 'Weak' (also you use a scientific term and an emotional term in the comparison) isn't huge. 2 HP per level (the difference between 10 and 15 CON) isn't that dramatic compared to a huge number of other factors (Class, positioning, not being surprised, Spell choice, Gear, enemy type, etc etc). I suspect What you mean is "optimal but boring" and "non-optimal but fun" where "optimal" means best at combat, and not best at other areas of the game.
4) Removing Con from the game does exactly the same as giving everyone 14 CON - therefore it solves absolutely nothing except piss off Dwarves, Goliaths and Half-orcs. In fact, just remove mechancial stats (Attack, spellcasting, Con) completely from the game and Max them dependent on class, and let people spend their Stat points purely on skills and ability checks? In fact, loads of systems already do exactly that, so why play D&D at all?
Look, I'll say it again, and again. If you believe maxing CON is integral to your enjoyment of the game, then that's fine. Firstly, there's plenty of Class/Race/Character Concepts that support this, and plenty of GM styles that accommodate. Great! You know what you want. However, claiming something is badly designed because it doesn't match your preference isn't fair. Especially, when I think it's brilliant that I can make a Fighter with 10 CON and still have great fun in all aspects of the game. The fact we can hold these contrary views shows exactly how well 5e is designed.
