The problem with keeping Gauthier as the "extra forward", is that you can't get him enough ice time in that role to play the "specialist" part he was ostensibly brought along for. You can't have Gauthier sitting there doing nothing for half a period, then toss him out at a critical moment for a 5 on 3 PK faceoff, or a defensive zone draw in the last minute of the game, one goal lead, extra attacker out...because he's going to be stone cold and completely out of the pace of the game (which is an issue for Gauthier to begin with).
And because Gauthier is so limited offensively, and not particularly versatile...you can't just sprinkle him in for a shift here and there with a bunch of different lines to keep him in the game, as you can with someone like Point for example who can play Center or Wing, on pretty much any line, without looking out of place. You're not going to sub the "13th forward" in for a shift on McDavid's Wing if your spare is Gauthier.
And Gauthier simply isn't the "best option" at this level, for matching up against opponent's top lines. So you've got a "checking line" that you don't really use to "check" top competition consistently.
With the "Men's Team" from the Olympics, our "checking specialists" were key players, guys who played a lot...Toews, Bergeron being the go-to guys for matchups. But also guys who don't look out of place if you want to say, give Bergeron a shift on the top-line wing with Crosby, or 4th line center or anything in between, to keep him in the game. Gauthier is not that.
Now that we're stuck with Gauthier though, i'm not really sure what you do with him, other than just have him spend the majority of the game sitting there twiddling his thumbs as insurance against an injury/misconduct/otherwise unavailable player...move somebody else up the lineup, and slot him in on the 4th line if you have to. Basically handicapping yourself to 12 forwards.
Or...just continue to play him as the "4th line Center" and live with the fact that the line isn't as effective as it could be, and hope that throwing that unit out there cold in a critical situation doesn't burn us too badly. Neither is a particularly attractive option to me.
That "defensive specialist" and "PK ace" role is just such a difficult one for most players to jump into fresh off the bench cold, out of the flow of the game, the legs not really going, etc. You're talking about basically the "toughest minutes" as an assignment, while only playing ~2 actual shifts per period? Disaster waiting to happen imo. It just doesn't work very well at this level, with players used to playing 20+ minutes a night.