“I don’t think they’d make the ambassador to the United States a KGB guy. It’s not really their style.”That is also how our own system works: the ambassador, in his role as Chief of Mission, leads and directs the Country Team. That team includes the CIA's Chief of Station. Though the COS reports back to Langley as well, the COM has the overall responsibility for directing US government operations in the country concerned.
That said, the job of any Russian ambassador is to oversee the rezidentura, or mission-bound spy station, putting Kislyak at the top of the pyramid of Russia’s security services in Washington. He would, therefore, likely have intimate knowledge of everything Russia's foreign and military intelligence operatives were up to in Washington and wouldn’t necessarily need to hail from the SVR or GRU or any of the other “power ministries" to cultivate assets and informants on foreign soil.
That said, American ambassadors are often political donors -- especially ambassadors to countries without much need for aggressive intelligence collection. Still, this is a major part of their role, as everyone knows who has to deal with ambassadors in any official capacity.
UPDATE: Right on cue, Wikileaks sets out to prove the point.