The thing that makes Logan's final words hurt more is the double meaning. I mean, yeah, you have the whole "here's a two hundred year old guy finally getting to feel death" but the other side of that is he loved but was never loved. His wife was a plant to keep an eye on him and Jean Grey never loved him either. So to see that little girl, crying for him as he died, meant he finally knew what being loved felt like.
With Stark, that final line is defiance. He isn't acknowledging his death, he's finally defeating his demon. His death doesn't actually start to hurt until Pepper is with him, telling him that everyone is going to be fine and he can rest now. Bonus points for Pete apologising yet again, emphasising how this kid never needed an Uncle Ben because he already felt the responsibility and the guilt of not having done enough.
I think what makes Logan work is that it doesn't feel at all like a comic book movie. I'm not even sure if you need to see any X-Men movies before it to understand what's going on, so universal is its story. There's a certain added layer of sadness to the whole movie as well when you consider Days of Future Past is set in 2023. Which means, sometime in the six years between Days of Future Past and Logan, everyone still dies. They just die a different way.
So Logan works on multiple levels, as a movie on its own merits and the finale to a seventeen year long saga.