The published landing distance for the Cessna is the certified distance - that's what it needs in a worst-case, no-flaps, high, hot, zero-wind landing. Any pilot will tell you that they rarely need the whole runway.
Consider the big picture of the aircraft carrier. The ocean is HUGE (ask the people searching for Malaysia 370) and rather featureless. Private planes rarely cross large paths of open ocean as it's often the longest route to where they are going rather than the shortest. So the chances of a carrier battle group being anywhere near you are extremely small. If your plane is flying well enough to find the CBG, come in behind and line up for a landing then you probably don't need it in the first place.
In theory, light aircraft would have no technical problems landing on the deck. Lots of room with a 30-50kt headwind, no need for a tailhook or nets as you don't weigh 30,000 pounds or come in at 200kt IAS on full afterburner. The chances of a non-navy-trained pilot hitting the touchdown zone of a runway moving away from them at 30kt and pitching up and down several meters is laughably small. You would either miss and have to go around (again, if you can do that, you don't need the carrier in the first place) or make a mess on the fantail. Even navy pilots do this on occasion, but they have ejection seats.