Starting on the back nine, Kuchar finished with another seven-under 63 on day two, opening with three straight birdies and an eagle on the 18th to go out in 30 before the turn.
The American followed up with two more birdies to move to 14 under overall and one shot clear of closest challenger Andrew Putnam, who carded a blemish-free five-under 65.
Former world number one Spieth, who has slipped to 17th in the world rankings, needed four birdies on the last four holes.
However he fell short of the cut by one shot after parring the 17th and narrowly missing an eagle putt on the last.
Scot Russel Knox carded a second-round three-under 67 to sit alongside day one leader Adam Svensson, who dropped down the leaderboard after a disappointing four-over 74
Chez Reavie, who holed eagles on three par-fours, and Stewart Cink are tied in third on 10-under, while Englishman Luke Donald also fell below the cut line and Ian Poulter fired a 69 for joint 65th place.