Nope...doesn't work at all.

Using "one" means everyone or anyone. The meaning is very different from the personal "I/me/mine". "One" is used in general terms.

For example...

I went to the store has a different meaning from one went to the store.

In your example, an experience cannot belong to one (meaning self), nor does it make any grammatical or inferential sense to say that "this is one's" when referring to oneself.

I see nothing wrong with using personal pronouns. Anything else seems precious, forced and, to me, silly. Language is good and using it correctly is even better. Reliable, accurate communication depends on using language well.