Conversation really can be a minefield. Between this thread, the count your blessings one, and whether or not to comfort someone crying, I am feeling a little overwhelmed!
I know that intent is not magical, but in some cases, I really think it has to count for something. A person's words are not always going to be the perfect thing, but perhaps we can give a little consideration to someone well meant?
To answer the question posed: no, I don't think blind (unconditional?) support is rude.
I do agree with you that intent matters, or rather, tone and context. If, in the course of a longer conversation about their specific concerns, you say to someone, "I know you'll win," I don't think that's going to crush them utterly. And, I think one can imagine how a "good" affirmation like "I know you'll do great" could be hurtful--picture it being said in a sarcastic way, or dismissively followed by, "But back to MY problems now." There's nothing magical about the specific words.
However, I like to learn how the words I say could be interpreted by (some) other people. For example, before eHell I used to say things like, "You look great now that you've lost weight!" and "When are you guys having kids?" as I heard other people say. I meant these things well and hopefully the person I was talking to knew this. But, I had never stopped to think what the implications of those words were--that the person looked bad before when they were fat, and that having children is the natural and desired next step. Again, not everyone "heard" those implications, and even among those who did, I hope most knew that I meant well. But, since it seems that a large number of people find those implications hurtful, I want to know this, so I can stop using those words. If "you will win" slips out of my mouth when talking to someone nervous, it's not the end of the world; but now I can articulate that this phrase has unhelpful implications, and I can make an informed decision to avoid that phrase in the future and substitute a different one in my attempt to be helpful.