As a vegetarian Muslim, I found a workaround by donating to a mosque that will slaughter the same animal whether or not I donate. This provides sufficient diffusion of responsibility for me to feel comfortable about killing an animal, while still participating in an important Muslim event.
I personally do not feel that kindness to animals is a character flaw. I think it's reasonable to be hesitant to participate in this:
(Image source: Daily Mail; link contains further graphic images)
As a whole, the impression I got from reading the whole Qur'an was that (some) animals were intended to be eaten by man. One could reasonably argue that eating meat is encouraged by the Qur'an. For example:
So eat of that [meat] upon which the name of Allah has been mentioned, if you are believers in His verses. And why should you not eat of that upon which the name of Allah has been mentioned while He has explained in detail to you what He has forbidden you, excepting that to which you are compelled. And indeed do many lead [others] astray through their [own] inclinations without knowledge. Indeed, your Lord - He is most knowing of the transgressors. Qur'an 6:118-119
Note that we also say bismillah ("in the name of God") over vegetarian food. Islamic Studies suggests the intention is to "recognize... as lawful all that God has declared to be lawful". Marwan Boustany similarly contests this is referring to meat literally, writing "This is not the case though, as Allah has not indicated this in these verses". Your mileage may vary as to whether or not this implies a command to eat meat or an assertion that it's okay to eat meat, or something else.
The Qur'an also has:
And the camels and cattle We have appointed for you as among the symbols of Allah; for you therein is good. So mention the name of Allah upon them when lined up [for sacrifice]; and when they are [lifeless] on their sides, then eat from them and feed the needy and the beggar. Thus have We subjected them to you that you may be grateful. -- Qur'an 22:36
It's hard to tell if this sacrifice should be interpreted as an obligation or not, but it's certainly not discouraged.
Some translations of Qur'an 21:107 include kindness to animals:
We sent thee not, but as a Mercy for all creatures.
while others do not.
We have sent you forth as a mercy to all mankind.
I've seen claims that these ayah support vegetarianism:
The seven heavens and the earth and whatever is in them exalt Him. And there is not a thing except that it exalts [ Allah ] by His praise, but you do not understand their [way of] exalting. Indeed, He is ever Forbearing and Forgiving. -- Qur'an 17:44
And there is no creature on [or within] the earth or bird that flies with its wings except [that they are] communities like you. We have not neglected in the Register a thing. Then unto their Lord they will be gathered. -- Qur'an 6:38
As far as I can tell, it's just wishful thinking to think they support vegetarianism.
While I'm not happy about it, I believe an honest reading of the Qur'an does not encourage being vegetarian, although it doesn't directly oppose being vegetarian. [As such, I'm conflicted about being vegetarian.]
In a way, I feel it's analogous to being invited over to someone's house, and when they offer you something to eat, you refuse. Well, we're in Allah's house, and he's offering meat.
The Prophet Muhammad was reported to have encouraged the slaughter during Eid:
"I attended (the day of) sacrifice with the Messenger of Allah He led the people 9in prayer, then when he finished praying he saw some sheep that had been sacrificed. He said 'Whoever slaughtered (his sacrifice) before the prayer. Let him slaughtered (his sacrifice) before the prayer, let him slaughter a sheep in its place, and whoever has not slaughtered, let him offer a sacrifice in the name of Allah, the Mighty and Sublime."' (Sunnah.com; Islam Q&A assert this was during Eid ul-Adha)
The Prophet Muhammad was reported to have personally sacrificed animals:
The Prophet sacrificed two horned rams which were white with black markings and had been castrated. (Sunnah.com)
The Prophet Muhammad was reported to have encouraged the sacrifice of animals:
The Companions of the Messenger of Allah said: ‘O Messenger of Allah, what are these sacrifices?’ He said: ‘The Sunnah of your father Ibrahim.’ They said: ‘What is there for us in them, O Messenger of Allah?’ He said: ‘For every hair, one merit.’ They said: ‘What about wool, O Messenger of Allah?’ He said: ‘For every hair of wool, one merit.’ (Sunnah.com)
At the same time, the Prophet Muhammad was reported to have prescribed relatively painless methods for slaughtering animals:
Verily Allah has prescribed ihsan (proficiency, perfection) in all things. So if you kill then kill well; and if you slaughter, then slaughter well. Let each one of you sharpen his blade and let him spare suffering to the animal he slaughters.” (Sunnah.com)
And the Prophet Muhammad is also noted for promoting kindness to animals, including sparing their lives:
He who takes pity even on a sparrow and spares its life, God will be merciful to him on the Day of Judgement. (Islam and Animals)
There is a reward for kindness to every living thing (Sunnah.com reference.)
As far as I know, scholars agree that vegetarianism and veganism is fine (provided you're aware that eating meat is halal [with the exclusions in the Qur'an]). Some examples sourced from Fatwas on Vegetarianism (more are found by following the link):
So traditionally Muslims were semi-vegetarians. The Prophet was, I mean, technically, the Prophet (SAWS) was in that category. He was not a meat-eater. Most of his meals did not have meat in them. And the proof of that is clearly in the Muwatta—when Sayyidina Umar says, ‘Beware of meat, because it has an addiction like the addiction of wine.’ -- Hamza Yusuf
A Muslim may be a vegetarian. However, he should not regard eating meat as prohibited. -- Mufti Ebrahim Desai
Vegetarianism is halal. Meat is not compulsory. Any food is permissible provided it is not harmful. Muslims are free to eat whatever they want provided it is halal. -- Sayyid Fadhlullah
The Salafi Islam Q&A say even veganism is okay:
I'm [vegan], I mean I do not eat... any [thing] that have to do with animal... can I be a Muslim and still [not] eat meat and animal [products]?
Yes, you can be a Muslim without eating these animal products.
You should not think that it is better to abstain from these foods, or that doing so will be rewarded, or that a vegetarian is closer to Allaah than others, and so on. -- Sheikh M. S. Al-Munajjid