I don't think that having a DPS check within an instance before allowing players to run certain content is a bad idea. It may seem offensive because we're in a social setting, but it is present in a lot of single-player games. Other games do it all the time. Fighting games wouldn't let you unlock a character you wanted to play as unless you defeated certain challenges. Single-player RPGs won't even let you access the main story that we can freely see in FFXIV unless you defeat grueling bosses.
Anyone who plays Shin Megami Tensei knows exactly what I'm talking about.
The hard truth is, that's just the name of the game when it comes to endgame content, and people that aren't putting forth the effort should realize that they're killing the fun for other people and, eventually, will kill it for themselves when they struggle to pass certain fights.
But, honestly, a DPS check wouldn't really solve the main issue. I honestly think the main issue here is that players simply don't respect each other. When I used to heal, I refused to level (arcanist?) for Swiftcast because, quite frankly, I didn't want to. And that was okay when we were running dungeons and nobody died, because I never needed to use it.
It was a different story altogether when we faced Twintania. I realized that there was simply no getting around it. I needed Swiftcast to raise people if I wanted to do the harder content in the game. It also came in handy when Stoneskin II was released.
Similarly, I don't currently have Blood for Blood to cross-class for MCH. So I'm working on that before I run harder content in Heavensward, but my parties do just fine in normal content without it.
We just have to recognize and respect what it is we're queuing up for and play the game accordingly. I know I wouldn't have fun adhering to a rigid DPS rotation because I, personally, would feel like a robot. And that's why I love MCH, because there is a rotation, but it's proc-based. To me, it's more organic and freeflowing than the other roles I tried out.
Just treat the game like alcohol. Have fun and play responsibly so that nobody gets hurt. This is a social, team-based game. Playing alone is fun, but you should always be able to have fun with the team you're playing with, and they should have fun running things with you. If not, then something is wrong.