I don't think I would be afraid to use the disavow tool in your case. You have a clean site so even if it triggered some kind of an audit, you would have nothing to worry about, correct?
I'm not worrying about an audit, I'm just worried that discarding 20,000+ inbound links from an apparently legitimate news aggregator might be counterproductive. The links were unsolicited and are on topic--they aren't obvious "negative SEO"--so it would make sense to leave them alone unless Google regards all outbound sidebar links with suspicion. (It's the "unless" part that makes me apprehensive.)