It is perfectly acceptable to have Fighters make social checks especially when the figther character takes the lead in a social encounter.
Okay, but if they fail and the party dies, then it's entirely your fault and everyone at the table will hold that against you forever. You
chose to kill the party, and waste months of time for everyone at the table, because you didn't want to let someone else take their turn in the spotlight after they've explicitly declared their intent ahead of time (by building a character that is competent at the task). You just
had to hog the spotlight, even though you
knew that you might fail. I can't imagine such an inconsiderate player would be welcome at any table for long.
Other PCs may assist. The DC need not always be 15+. The Fighter is permitted to increase his/her Charisma.
A fighter who increases their Charisma instead of a useful stat is a liability to the party, and they're going to get everyone killed. Don't be that player, who puts their own character quirks ahead of their responsibility as part of the team. Either build a functional character who is competent at their job, or go play a video game so you're not dragging down everyone else.
And even if you
do have some selfish, self-absorbed player who increases their Charisma up to 16 (because they don't care who else suffers from their poor choices), they will still
never be able to reliably hit DC 11. I don't know what kind of game you're running where anything that really matters to a level 17+ character will still be hinging on a DC 10 check.
One may select a background/feat to allow one to pursue/assist in the social pillar. A Fighter may use their inspiration. Depending on the fighter's actions/words/player innovativeness they might be awared with a bonus or even advantage.
Sure, you can always invoke obscure optional rules, or make up new rules of your own, to address shortcomings within the system. That doesn't excuse the system, itself, for being faulty.