As a DM - I try to encourage players to get at least one 16 on their PC one way or another as I find that even with the best and noblest of intentions, players lose focus more quickly when other PCs outshine theirs. There are exceptions to the rule - but it is an overwhelmingly strong tendency.
People fantasize about being powerful. They rarely fantasize about being weaker. You can tell great stories about weaker characters, but in the end playing one for 800 hours is different than reading about one or watching one on screen for a bit.
When players stick to their guns, I account for their weakness and give them an option for a backdoor to power. It might be as simple as a Headband of intellect or Gauntlets of Ogre Power ... or it might be giving them a storyline that puts a powerful magic item in their hands early. Again: This gives them an optional out on their initial preference.
Is this a DM saying, "I know better than the players and am building in an option to let them change their mind because I think they are likely to do so?" Yes. This is exactly that - a DM making sure that he preserves paths to fun that the player might appreciate more later, but does not have to take. It is DM care of players.
As a player - After I play a strong PC I like to play a weaker PC. It isn't always about attributes. Sometimes it is just being the PC that buffs, heals, absorbs hits and solves puzzles while the other PCs beat the @%@ out of the foes. However, sometimes it is being Robin to another player's Batman, or Sancho Panza to another player's Don Quixote with a 12 or 14 strength Champion fighter.
I like the switch because it gives me a different story to play.
Ah, but doesn't this 100% invalidate your assumption as a DM that players need to be provided an out if they elect to play low attribute PCs? Not really. Because there have been times that I have started out with a weak character that filled a utility role and it got boring because there wasn't much I could do. In those combat focused games, my utility role was ... meaningless. And that was a situation in which I made the exact wrong call to play a weaker PC because in that game, it did not work. DMs sometimes gave me a fix (especially if I hinted one would be appreciated), but other times my PC ended up leaning towards the high danger play in order to 'pull his weight'... and overextending your PC in D&D can be quickly deadly.
