The scar, the pain it causes sometimes, the bad dreams and the little trips inside Voldemort's mind, and the ability to talk to snakes are all quite unpleasant things in themselves to live with.
“He’s a Parselmouth.
Everyone knows that’s the mark of a Dark wizard. Have you ever
heard of a decent one who could talk to snakes? They called
Slytherin himself Serpent-tongue."
(Chamber of secrets, Chapter 11, The Duelling Club)
Harry lay flat on his back, breathing hard as though he had been running. He had awoken from a vivid dream with his hands pressed over his face. The old scar on his forehead, which was shaped like a bolt of lightning, was burning beneath his fingers as though someone had just pressed a white-hot wire to his skin.
(Goblet of Fire, Chapter 2, The scar)
And as he thought this, the scar on his forehead burned so badly
that he clapped his hand to it.
“What’s up?” said Hermione, looking alarmed.
“Scar,” Harry mumbled. “But it’s nothing...It happens all the
(Order of the Phoenix, Chapter 9, The Woes of Mrs Weasley)
His restless night was punctuated once more by dreams
of long corridors and locked doors, and he awoke next day with his
scar prickling again.
(Order of the Phoenix, Chapter 15, The Hogwarts High Inquisitor)
He looked around at the
others. They were all staring at him, the strings still trailing from their
ears, looking suddenly fearful.
(Order of the Phoenix, Chapter 22, St Mungo's Hospital)
He felt dirty, contaminated, as though he were carrying some
deadly germ, unworthy to sit on the underground train back from the
hospital with innocent, clean people whose minds and bodies were
free of the taint of Voldemort...
(Order of the Phoenix, Chapter 23, Christmas on the Closed Ward)
But, most importantly, the scar, the pain it causes sometimes, the bad dreams and the little trips inside Voldemort's mind, and the ability to talk to snakes are all symptoms of the fact that Harry has in him a part of someone else.
That's something already quite unpleasing in itself if you ask me, but it turns out that "someone else" is also the darkest wizard of the time, the guy who killed his parents and who tries repeatedly to kill him.
Oh, and his god-father and many of his friends also died because of this "someone else" (and for Sirius, almost directly because of that stranger part of soul being in him).
“You can speak Parseltongue, Harry,” said Dumbledore calmly,
“because Lord Voldemort — who is the last remaining descendant
of Salazar Slytherin — can speak Parseltongue. Unless I’m much
mistaken, he transferred some of his own powers to you the night
he gave you that scar. Not something he intended to do, I’m
“Voldemort put a bit of himself in me?” Harry said, thunderstruck.
“It certainly seems so.”
(Chamber of secrets, Chapter 18, Dobby’s reward)
I guess what I'm trying to say is that Harry does not like Voldemort, and has every reason to very much dislike hosting a part of him.
And so, it's completey normal that he's perfectly happy to be rid of this burden and to only host his own soul.
I think it explains why Harry is happy to be rid of each of these symptoms, including the Parseltongue thing.
The scar had not pained Harry for nineteen years. All was well.
(Deathly Hallows, Chapter 37, Epilogue - Ninteeen years later)
Little disgression as a PS:
The only thing he loves that links him to Voldemort is his wand... but well he lived so much with it, and this wand has helped him so much, including when fighting against Voldemort.. I think the "this wand links me to Fawkes and Dumbledore" winning over the "this wand links me to Voldemort" is really well deserved.