It's a combination of things-
1) Betts is not a flashy dude who needs the limelight. He appears to genuinely like the Washington area and has a good relationship with the team, which drafted him during the Spurrier years and has kept him into now the third regime. He has a role that he's comfortable with.
2) The league is saturated with RB talent right now - more than I can ever remember. Almost all teams are two-deep with RB's who could put up 4 ypc and 1000 yards with good enough talent around them. Good-but-not-great former backups who go elsewhere to try to start haven't really done all that well. Are you impressed with how Lamont Jordan's career took off after he left the Jets? Reuben Droughns? Tatum Bell? Mike Anderson? Michael Bennett? Chester Taylor? Chris Brown? Dominic Rhodes? If people are predicting great things for Julius Jones this year, I'm not seeing it. Yes, you get occasional outrageous success stories like Priest Holmes, but typically a "success story" in this scenario is a guy who can put together intermittent starting roles here and there as he moves teams, like you've seen with Thomas Jones or Michael Pittman.
The problem is that you're good-but-not-great, and everyone knows that. Under the right scenario, you're going to get replaced like you saw with Chester Taylor and Adrian Peterson, or Thomas Jones with Cedric Benson. Honestly, as successful as Ryan Grant has been, if that team goes 7-9 and is picking at 1.10 next year, and is staring a playmaking RB prospect in the face with that pick do you really think they're going to avoid drafting that RB because Ryan Grant is just too good to upgrade?
Anyway, I think Betts took a look around and realistically assessed his prospects, and thought it was better to be paid well as a backup RB for a team and in a place that he and his family liked rather than chasing something that he probably couldn't keep for very long even if he caught it to begin with. While there's a time and a place for blind ambition, I admire what I take to be the maturity in his decision above and beyond being thankful for the decision as a Redskins fan.