I've been to Asheville dozens of times and never bothered with the Biltmore. I've toured lesser homes elsewhere and paid much less to do so. A great way to see it without going in is to buy or rent the movie "Being There". Besides having been filmed on site, it's a great movie and Peter Sellers' swan song.
Some of my kids have been on school trips. I don't recall paying anything like $60 and lunch and the bus from Knoxville would have been included. Any way you can get in with a group?
My favorite activity (excluding restaurants and breweries) in Asheville is the zoo. It's very convenient, well laid out, well stocked with native wildlife, and *this can be key* well shaded.
Be careful with your accommodations. I once used Priceline or Hotwire and ended up in an absolute dump right at that exit off I-40.