I don't believe so, at least my '67 doesn't re: the first gear start. It should start in second unless it's almost floored or the gear shifter is all the way forward into "2".

Your car has a switch on the front carb which controls an important servo on the side of the transmission. With the engine off and the ignition ON, you can move the throttle a hair under the hood and hear the servo make a loud click. If it doesn't click, then that's probably the cause of your stuttering, it may be stuck in "idle/decel" mode. There are three positions to the servo: kickdown for full throttle, normal driving, and idle. It is external, you can see it move on the side of the tranny. (Similarly you should hear it click when the throttle is floored as well)

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1967 230S automatic
Boston, MA