Did the Enterprise See Herself?

While writing on a page, a thought crossed my mind.

You see, it seems very odd that the Federation would have authorized the intentional time travel mission from "Assignment: Earth".   Even if you assume that the timeline is pretty robust and resilient, it doesn't seem wise to plop an entire bright-white 289 meter Constitution Class starship in 1968 Earth orbit when the same job could be achieved by a specialized low-observability vessel or microsatellite or somesuch.

However, it also occurs to me that "Tomorrow is Yesterday" did feature an accidental dropping of the same Constitution Class starship into Earth's atmosphere.   On that occasion the Enterprise escaped via some sort of unusually slow time-warp maneuver.  Her sensors were capable of recording information at the time, which is how they were able to do the beaming maneuver as they slowly returned to their time.

So, as they travelled forward in time with sensors recording information, did the Enterprise see herself in 1968, which led Starfleet's eggheads who were later poring through the sensor logs to send them back there?   Was her assignment to 1968 Earth thus a predestination paradox?

Just a random thought.