Oh, there is a carrier out there. And various aircraft, with known signatures routinely take off and land from it all day long. We even track many of the ID numbers.
There are a number of tracking sites, although normally don’t get the actual fighters, but can pick up other aircraft and re-fuelers for those.
Historical context is important.
The problem is, we have cyclically (off and on), had a carrier out there, (positioned or along), The Southeast and Florida coast (often related to NAS Jacksonville), for decades. It’s not constant in time, but it is normal.
I'm assuming you're talking about the Training Carriers (aka AVT's) used for Student Naval Aviators and normally stationed off JAX or more frequently off the Texas Gulf Coast where the advanced training bases were located.
In my day, AVT-16 Lexington was used. "Lex" got replaced by the Forestal and is now on display at Corpus Christi.
Forestal did her turn until the early 90's when decommissioned and later scrapped.
Since then, the Navy hasn't had a designated AVT, preferring to train Student pilots their carrier landing skills in simulators.
I suppose the occasional CVAN may slide down by JAX for old times sake but I'm afraid the times when the Lex or Forestal were there ended 30 years ago.
So if there's a CVAN (supposedly with the rest of the Battle Group) or something else requiring the services of a Greyhound out there, it ain't standard by any means.