First, I sympathize with the argument that experience is primordial. However, you seem to assume that before an office term, the politician was not assuming any other public function. Likewise, you seem to assume that after the term, due to term-limit, the politician leaves the public administration in its entirety. Experience could be built in other offices/functions, so I don't really understand your point.
Second, there has been an interesting couple of papers written by Kroszner and Stratmann(2000, 2006) about the committee structure in Congress as a way to reduce an imperfect information problem.
The idea is that Political Action Committees cannot pay legislators to represent them. If they could, they would. Because such contracts are not possible/enforceable, the legislator that gets paid by the PAC(e.g. for campaign financing) has no incentive whatsoever to actually defend the interest of the PAC as soon as it gets the money. Likewise, the PAC has no incentive to pay the legislator if the guy defended the PAC interest on one law.
However, we have a repeated game: congressmen can indeed engage in reputation building. Here is why I think it is linked to your post: one problem with term limits is that you hamper reputation building.
One good thing with reputation building is that you correct the information imperfection: if the legislator does not satisfy the PAC requirement once he has been paid, he can't expect the PAC to pay for his/her campaign in the future.
The committee system is good in this sense because it gives more information on the field of interest/influence of a legislator, and also because usually legislators stand in the same committee over time.
Those papers have been written for congress, but I guess some kind of reputation-building might be at stake with governors and state legislatures too