WARNING: Spoilers ahead.
(Emphasis mine on all quotes below)
Why was Hedwig killed anyway?
Other than the twitter link provided in the question, Rowling gave a fuller explanation in an interview (full transcript of the interview can be found here):
Twinkletoes*: "Why did you feel that Hedwig's death was necessary?"
J.K. Rowling: "The loss of Hedwig represented a loss of innocence and security. She has been almost like a cuddly toy to Harry at times. Voldemort killing her marked the end of childhood. I’m sorry… I know that death upset a LOT of people!"
* Username of person asking question
Hedwig was the last remaining part of his childhood. Ron and Hermione both had grown up, and this supposedly caused Harry to become an adult.
The name 'Hedwig'
'Hedwig' symbolises other things too. St. Hedwig was the patron saint of orphans, making the owl even more special to Harry, who was himself an orphan:
He had had it as long as he could remember, and the first question he could ever remember asking his Aunt Petunia was how he had gotten it [the scar].
"In the car crash when your parents died," she had said. "And don't ask
questions."
-- Harry Potter and the Philosopher's Stone - Chapter 2
Harry also chooses the name 'Hedwig' from one of his schoolbooks:
Harry kept to his room, with his new owl for company. He had decided to
call her Hedwig, a name he had found in A History of Magic. His school
books were very interesting. He lay on his bed reading late into the
night, Hedwig swooping in and out of the open window as she pleased.
-- Harry Potter and the Philosopher's Stone - Chapter 6
The fact that the name came from his schoolbook, suggests that with the death of Hedwig, his schooldays and childhood was over along with his innocence.
Colouring
Hedwig in the book was white, a symbol of innocence and the protection of other, saint like, people. When she was killed this innocence was gone.
The purity and innocence of Harry when he is a child is represented by the colour white, and Hedwig is a snowy owl, the whitest owl there is.
Harry now carried a large cage that held a beautiful snowy owl, fast asleep with her head under her wing.
-- Harry Potter and the Philosopher's Stone - Chapter 5
Friends
Back at the Dursleys', Hedwig was also basically Harry's only childhood friend outside of Hogwarts and magic. This is summed up perfectly in this extract:
In addition to every other miserable feeling, he now felt guilty
that he’d been irritable with Hedwig; she was the only friend he had at
number four, Privet Drive.
-- Harry Potter and the Order of the Phoenix - Chapter 3
And perhaps even better, this quote from when Harry has just lost Hedwig:
"Wait a moment," said Hagrid, looking around. "Harry, where’s
Hedwig?"
"She . . . she got hit," said Harry.
The realization crashed over him: He felt ashamed of himself as
the tears stung his eyes. The owl had been his companion, his one
great link with the magical world whenever he had been forced to
return to the Dursleys.
He misses her when she is gone:
Hedwig had been absent for two nights now. Harry wasn’t worried about her: she’d been gone this long before. But he hoped she’d be back soon — she was the only living creature in this house who didn’t flinch at the sight of him.
-- Harry Potter and the Prisoner of Azkaban - Chapter 1
They have a mutual friendship:
She, too, was carrying a parcel and looked extremely pleased with herself. She gave Harry an affectionate nip with her beak as he removed her burden, then flew across the room to join Errol.
-- Harry Potter and the Prisoner of Azkaban - Chapter 1
The snowy owl clicked her beak and nibbled his ear affectionately as Harry stroked her feathers.
-- Harry Potter and the Order of the Phoenix - Chapter 4
He walked away from them, looking around, with Hedwig nestled contentedly on his shoulder, but this room was not likely to raise his spirits
-- Harry Potter and the Order of the Phoenix - Chapter 4
Hedwig hooted happily at Harry from her perch on top of a large wardrobe, then took off through the window; Harry knew she had been waiting to see him before going hunting.
-- Harry Potter and the Half Blood Prince - Chapter 5
Her death shows once again how the remnants of Harry's childhood were gone.
Birthdays
Credit to Torisuda for posting a comment that got me started on this bit.
While Dudley is always getting a ton of birthday presents:
Dudley, meanwhile, was counting his presents. His face fell.
"Thirty-six," he said, looking up at his mother and father. "That's two
less than last year."
-- Harry Potter and the Philosopher's Stone - Chapter 2
All Harry got for his tenth birthday was a coat hanger and a pair of socks:
...then tomorrow, Tuesday, was Harry's eleventh birthday. Of
course, his birthdays were never exactly fun -- last year, the Dursleys
had given him a coat hanger and a pair of Uncle Vernon's old socks.
Still, you weren't eleven every day.
-- Harry Potter and the Philosopher's Stone - Chapter 3
So when Hagrid gets Harry his first true birthday present, on his first birthday as a wizard, he instantly becomes attached:
"Just yer wand left - A yeah, an' I still haven't got yeh a birthday
present."
Harry felt himself go red.
"You don't have to --"
"I know I don't have to. Tell yeh what, I'll get yer animal. Not a toad,
toads went outta fashion years ago, yeh'd be laughed at - an' I don'
like cats, they make me sneeze. I'll get yer an owl. All the kids want owls, they're dead useful, carry yer mail an' everythin'."
Twenty minutes later, they left Eeylops Owl Emporium, which had been
dark and full of rustling and flickering, jewel-bright eyes. Harry now
carried a large cage that held a beautiful snowy owl, fast asleep with
her head under her wing. He couldn't stop stammering his thanks,
sounding just like Professor Quirrell.
"Don' mention it," said Hagrid gruffly. "Don' expect you've had a lotta
presents from them Dursleys.
-- Harry Potter and the Philosopher's Stone - Chapter 5
And then a couple of weeks before the day of his seventeenth birthday, his first birthday as an adult wizard:
[Mrs Weasley] Actually, I’ve been wanting to ask you how you want to celebrate your birthday, Harry. Seventeen, after all, it’s an important day. . . .”
-- Harry Potter and the Deathly Hallows - Chapter 6
Hedwig dies, leaving Harry distraught:
A second’s relief, and then another burst of green light. The owl screeched and fell to the floor of the cage.
“No — NO!”
The motorbike zoomed forward; Harry glimpsed hooded Death Eaters scattering as Hagrid blasted through their circle.
“Hedwig — Hedwig —”
But the owl lay motionless and pathetic as a toy on the floor of
her cage. He could not take it in, and his terror for the others was
paramount.
-- Harry Potter and the Deathly Hallows - Chapter 4
It seems more than just a coincidence that Harry received Hedwig on his first wizard birthday, in the first book, and lost her on just before his first adult wizard birthday, in the last book. She was present throughout his wizarding childhood, and died just before he left it.
Children's Toys
It has been a recurring theme that Hedwig was almost a child's toy to Harry.
Rowling mentions it in the aforementioned interview:
She has been almost like a cuddly toy to Harry at times.
And it is mentioned once again when she dies in the book.
But the owl lay motionless and pathetic as a toy on the floor of
her cage.
-- Harry Potter and the Deathly Hallows - Chapter 4
This 'cuddly toy' idea, attaches Hedwig in a special way to Harry. She was his pet, his toy, his friend, and so much else.
I hope this provides an answer, there is a lot of evidence, you just have to take a second look.
How this was done:
I developed this answer over a few days, mostly because I didn't have the time to do it all at once. First I checked through the first and last books, before checking the rest.
To search through a book, I would open up a pdf of it and then search through it using Ctrl+F. I searched 'Hedwig' in each and came up with the quotes above. I obviously added the emphasis myself. After a couple of edits, it is the answer you know see before you.
