First, you have to accept that, at least for the Tenth Doctor, regeneration is death. Stop thinking of other regeneration scenes and think of the process itself. We can see the Doctor is a different person each time. The other characters also form a different relationships with him. His physical appearance, personality, morals, feelings, everything we associate with a person changes. The only things that remain are memories and some basic features, like his will to travel. For some people, it's enough to come to terms he's the same person even when he's not. Now, Time Lords are normally less disturbed with regeneration than humans, as it's natural for them and not for us. But - Tenth Doctor was more human than his other versions. He had more human feelings and needs
Now I want to say why I love the "I don't want to go" line so much. We have no doubt the Doctor doesn't regret saving Wilf; he wasn't forced to do it and he'd do it again. He knows his death had a purpose. Then he accepted his fate. He had a chance to say goodbye to his friends and he did a final favour for each of them. He had everything you could wish for death, at least in storytelling, right? He should regenerate in peace. But - I don't think it works that way. You might be heroic and selfless, you might make brave decisions and accept universal laws, but at the end of the day, you still cling to life. It's instinct, we want to live. There's no way to know if there's anything else after death.
And the Tenth Doctor had a complicated relationship with life. With Rose, his sheer existence made him happy. Yes, he had darker moments, but you could tell he was a man who loved being alive, even if he was somewhat delusional about his future with Rose. Then things went worse and worse. He was close to suicide once; he shouted to the Daleks to kill him; he told Jackson Lake he had no one to live for; he was ready to sacrifice his life several times and only other people's sacrifices saved him. Of course, between those things, he could still enjoy a trip to a different planet, he'd still show off to his companions. But dying wasn't an abstract thought for him and there were moments in his life when dying would be less tragic and more like taking pain and burden of his shoulders. He found peace, we would say on those occasions.
But if the Tenth Doctor regenerated that way - in peace - it would mean that he lost. That he stopped struggling and fighting for life, and gave up. It would be so much more depressing than his despair, because it would mean he was permanently broken and destroyed, and if the Doctor's will to live, that basic instinct, was destroyed, well, what chances do we stand? I find it comforting that the Doctor, despite having a meaningful, but hard life, a purposeful death and time to see his friends for the last time still wants to live.