Ok I got to admit that I don't quite get it even after reading it a few times.

Lets start from the beginning: "Why bother with philosophy when it doesn't help you live?" - sounds like "Why bother with exercising when it doesn't help you live?" and I happen to agree with this Pierre Hadot that you mentioned.

My current understanding of philosophy is that it's like building. You start from somewhere solid and then you build your way up. And maybe demolish it if it shakes too much.

What is truth seems like a good start. And that's where I got lost. That quote from Alain Badiou and even your explanation. Here are some of the things I didn't get:

"Badiou is interested in events, in radical moments in which a subject (person) is completely changed." - I think you refer to the mind that is completely changed and not the body. So does that mean that the person is the mind?

"Truth is a state of exception, something we desire and which we try to find." - Ok, we do desire and try to find truth, but isn't that the case only when we arent in a state of belief(if I belive I know I dont try to find the truth. Why whould I?). And then "Truth is a state of exception" - I dont know what you mean. What kind of exception? Here is where I got completely lost haha.

"But our thoughts and our behaviour cloud reality, make it difficult to reach for this event, this happening of truth." - Doesnt that mean that in order to reach that event of truth we need to change our behaviour in some way and stop our thoughts? Stoping thoughts doesnt sound like a good idea. Maybe not paying them attention? Anyway, I'm not agreeing or disagreeing, just trying to clarify the meaning.

"When we walk on the sidewalk and a car passes us in the street, we are not amazed that we were not hit by that large metal thing on four wheels." - Werent we amazed by things like that when we were kids? So when you say "Well, we are all part of a world, we act according to its rules and regulations, and normally we don't even question the habits that belong to this world." - Does that mean kids dont act acording to the world's rules and regulations. Arent those rules and regulations just our own conclusions?

"It's great that things work like that, because if they didn't, we wouldn't make it out of our beds every morning." - Hmm, wouldnt that be even one more reason to get out of bed and explore?

Ok I'll stop here since the comment is getting kind of big and I don't want to annoy you too much ahaha.