I haven't actually seen the Bourne movies, so I can't pass judgement on them, or whether I think he makes a decent action star. Look back to when Keanu was making the Bill & Ted movies (which are awesome, by the way!) - who'd have ever thought he would emerge in the 90s as an action star? Who expected Will Smith to go from The Fresh Prince to appearing in Bad Boys and Independence Day?! A certain actor may not look like your typical action star, but as the above examples demonstrate, appearances can be deceiving.
The beefed-up hulks of the 80s (Arnie, Sly, Dolph, etc) gave way in the 90s to the likes of Keanu and Will, but as Dolphage pointed out, people are starting to want to see a return to the testosterone-fuelled one-man armies of yesteryear. Sly seems to have more or less kickstarted this by resurrecting Rocky and Rambo, and with a new Rambo on the way, and with Dolph bringing out multiple new films over the next couple of years, it looks as though those people are in for a treat!