As Tol so succinctly put it, lived experiences are hard evidence. So is a murder victim's body, which gives mute testimony of a lived experience so intense, it became a death experience. Otherwise, there would be no such thing as witness testimony and presentation of physical evidence in court. So, let me just re-name lived experience as witness testimony.
Contrast this with "witnessing" for Jesus is, i.e., people expressing their emotions to others in context of their experiences and beliefs. Although I become bored and annoyed when a Christian "testifies" or "witnesses" to me for more than ten seconds, I would not be so disrespectful as to call them out for being brainwashed. I would just make an excuse to leave. They are just gushing with happiness like an innocent, deluded child. There are no hard facts being discussed, just emotional reactions to coincidence and hyper application of pattern recognition.
However, if some woman described her lived experiences of male domination, such as being constantly interrupted by a douche bag during board meetings, but you don't like her choice of words or angry tone, maybe just focus on the particulars, the facts in the testimony. Although she is not testifying in a court of law, she is not in a debate contest, she has not spoken where she should expect her testimony to be judged as worthy of your approval, her testimony could nevertheless be admitted in court. She is describing actual events that really happened in real time, which others present at that same event could corroborate. Unlike someone witnessing their Happy Jesus Feels because gawdamighty made the bus run late, just so they could made it to work on time.
Everybody ultimately exists as a thinking, feeling entity inside their own head. Sometimes opening a window into that mental reality can expose some harsh or disturbing imagery. Life can be very harsh and disturbing. If you had a harsh and disturbing experience and expressed your feelings about it, I seriously doubt you would tolerate being judged by strangers as to the tone or word choice or on the facts of the incident you lived through.