I don't think it helped that Kathie Lee was just plain exhausting. I watched too much of Live in those years, mostly when staying with loved ones who had it on in the background every morning, or when in doctor's offices, and the shift from a "perky" figure to one who would spend 20-30 minutes every episode venting and complaining and being aggrieved was suffocating, even if I did share her views of the tabloids. Kathie Lee has never really been able to see the world around her, which later led to unfortunate moments like asking Martin Short questions about his happy marriage without knowing his wife had been dead for several years.
The tabloids had lost some of their power by the time Kelly came along, but Kelly also just wasn't that interesting to write about in comparison. She's mostly a quietly tiring, shut down presence.