The most genuinely good person I've known, my wife's very devout grandmother, who belonged to the United Church of Christ and whose husband was a preacher for a United Methodist Church, never once went out of her way to proselytize (in my presence at least.) That's not to say she wouldn't talk about God or her beliefs if it was somehow directly relevant to a conversation, but I never saw her trying to "spread the word" just to do so. She emphasized caring for others and the "love your neighbor as yourself," part. Her favorite scripture was the story of the Good Samaritan and that was what she requested to be read at her funeral. She didn't spend time talking about what Jesus wanted, or trying to convince you to believe, she set the example by living it.
The unfortunate part is, in my 38 years of existence, she is possibly the first "real" Christian I've known: Christian through her acts, not just her words. In the Parable of the Sheep and the Goats, Jesus bases judgment not on religious rituals or verbal professions of faith, but on acts of compassion towards others: feeding the hungry, giving drink to the thirsty, welcoming the stranger, clothing the unclothed, caring for the sick, and visiting the imprisoned. This seems to emphasize practical love and concern for the vulnerable as true evidence of discipleship, rather than one's ability to proselytize.
Isn't the most important part of the New Testament the Gospels and almost everything that comes after is more about institutional Christianity? Acts, describing the spread of Christianity after Jesus' ascension, with little new information about his teachings. Epistles focus primarily on doctrine, ethics, and church organization rather than Jesus' words and actions. Revelations...well I'll leave that to others to interpret.
If Christianity is built around Jesus, shouldn't his teachings and words be the focus rather than the institution of Christianity itself? I don't think there is anything wrong with proselytizing per se, it's obviously critical to ensuring the survival and expansion of any belief system and from an evolutionary standpoint, ideas that encourage their own reproduction tend to thrive. But it seems misguided to focus on spreading the faith rather than doing all the things Jesus says you should do.