The Buddha gave us indications as to what kinds of utterances or thoughts arise when Nibbana is reached.
Here, Ānanda, a bhikkhu thinks thus: ‘This is peaceful, this is sublime, that is, the stilling of all activities, the relinquishing of all acquisitions, the destruction of craving, dispassion, cessation, nibbāna.’
AFAIKT - this isn’t a literary device. Bhikkhus who reach Nibbana literally think these things when they arrive.
And, you’re right. Nibbana is an insight achieved after dwelling in the fourth jhana. And, moreover, it would be, AFAIKT, the case also that one who achieves first jhana notes to themselves:
“This is rapture and bliss accompanied by applied and sustained thought.”
Or:
“Applied and sustained thought have faded. There is only rapture and bliss.”
And so forth. Moreover, the gradual enlightenment taught by the Buddha is so well formulated and taught that, upon reaching forth jhana, the following utterance applies:
In the same way—with his mind thus concentrated, purified and bright, unblemished, free from defects, pliant, malleable, steady and attained to imperturbability—the monk directs and inclines it to the knowledge of the ending of the mental fermentations
So these are the stages a person will go through and the milestones they’ll reach. There will be no doubt in the mind that one has reached the goal and that there is no further searching to be done.
However much people vacillate over the meaning and nature of the goal, the teachings of the Buddha are succinct. There is no more and no less to add to them. And they guide the student along the path every step of the way.