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The NFL is changing shape. We've seen it for years, with Tom Brady and Peyton Manning (and Kurt Warner) leading pass-heavy offenses to the Super Bowl, flying in the face of the common (or old) philosophy that championships are won with a team's defense and running attack.
The San Diego Chargers have an elite QB, a future Hall of Famer at TE, a great one-on-one WR, and one of the best pass-catching RBs in the entire league. Do they really need to replace Ryan Mathews? Do they really need a "bellcow" RB to carry the ball 20+ times per game?
Consider the fact that NFL teams always look to the Super Bowl winner as a model to copy, and you could see how the Chargers might talk themselves into something totally different than what they've been trying. The New England Patriots went 3-0 in the playoffs, in games when their leading rushing had 6 carries (Tom Brady), 30 carries (LeGarrette Blount) and 14 carries (Blount).
Blount just narrowly edged out Jonas Gray for "random guy the Patriots decide to give the ball to when the opposing team can't stop the run." Neither guy gets paid much of anything, and neither were drafted by the Patriots. If they were unavailable, I don't think Bill Belichick or Josh McDaniels would've cared who replaced them, as long as it was someone that didn't fumble when he got hit.
I believe this is how the Chargers should address the hole left in their offense by Ryan Mathews' contract running out. Build up the offensive line, use Danny Woodhead as much as possible, and hand the ball to (random name) when facing a team that is losing the battle on the line of scrimmage so that he can pick up his easy 4 yards per carry. If (random name) asks for more money, let him go somewhere else (like the Pittsburgh Steelers, before he throws a fit and gets released and comes crawling back for the veteran minimum). If he is late to a team meeting, send him to Siberia.
Ryan Mathews is a special player, but he's not special enough. He has flaws, like health and occasional fumblitis. It would make way more sense to just let Donald Brown be the (random name) player that carries the "load" for the offense, and to switch the focus on short-yardage situations to short passes instead of runs up the gut.
Invite a bunch of undrafted free agents to camp and let one of them win that roster spot. Hell, maybe he can start if he's better than Donald Brown at staying healthy and not fumbling. However, never value the running back position as anything more than a nameless, faceless football player that is easily replaceable. Just like the Super Bowl Champions do.