Congress can certainly do more than one thing at once, but the media generally can't, so this becomes, more than anything, a question about whether having Trump at the center of the news daily will hurt or help him. History shows it may well help him, although his crimes are so much more egregious than Clinton's that it may also peel away a few of the Jeff Flake/Bob Corker type Republicans. On the other hand, he's probably already lost those Republicans.
And Trump will probably figure out a way to put himself at the center of the news cycle anyway, from now through election day. So, the real question is: Should impeachment hearings begin now, or after we hear from Mueller?