Given the fact that they were unaware of the barrier
Why "given"?
Our heroes are headed out of the galaxy, and they first make an unscheduled stop to pick up the recorder marker, but then make a scheduled stop at the "doorstep", where the barrier is clearly visible. Nobody seems to be surprised to find it there. Kirk then orders the ship to head out, at which point his crew begins an analysis of this "forcefield of some kind" - as if this were the ship's designated mission.
And indeed Kirk appears to deliberately fly into a phenomenon he could steer around, explaining that he has to assess the dangers so that others can follow.
For all we know, Kirk's mission was a success and all the goals were met: the risks of leaving the galaxy were established. Later on, leaving the galaxy is portrayed as a fairly mundane event, in the sense that it's considered risky (except the risks fail to materialize in "By Any Other Name" and are different in "Is There In Truth No Beauty?" from the ones in "Where No Man Has Gone Before") but not unique or exceptional in itself. Possibly others read Kirk's report, prepared accordingly, and probed into the space beyond the barrier, experiencing standard Star Trek adventures there.
Kirk's pilot mission might be part of an initiative to probe beyond the barrier, or then just a preliminary sortie that will some day allow Starfleet to propose such an initiative to the funding parties. We know there's some sort of infrastructure in place, including a refinery in the immediate proximity of Kirk's departure/reentry point. It's apparently not yet an integral part of Kirk's mission, as Spock has to remind Kirk of its existence when it promises to be relief for their acute crisis. But it's probably not a coincidence that Kirk chose to depart the galaxy at that specific point...
Incidentally, there's no indication that the barrier would block anybody's view of the outside. Starfleet might be fully aware of the interesting star systems that lie beyond this well-defined barrier, and eager to explore them with starships (because Star Trek nicely establishes that remote observations don't reveal much in that universe, and entire planets can go missing until a starship visit establishes their current status). The inside and outside of the barrier might not be exotically different, not until one sails a further couple of thousand lightyears from the barrier and reaches the halo. But the barrier itself would be an intriguing, "population-separating" phenomenon, worth crossing for the very reason that it is there.
Timo Saloniemi