What you are referring to is a Rule 68 (FED. R. Civ. P. 68) offer of judgment (OOJ). Thus far, nearly every jurisdiction's court of appeals has refused to construe these as being binding on the Plaintiff if the offer is denied, even if it offers complete remuneration, especially in a putative class action. The underlying reasons when applied to a certified class differ fairly substantially from an individual plaintiff or a non-certified class. However, there are some kinds of cases where a Rule 68 offer could never fully compensate (as with cases where subjective or non-substantive forms of damage have been requested).
A Rule 68 offer is a cost-flipping mechanism, often used by defense attorneys when they are making what they believe is a fair offer (very rarely is the offer one that is equal to the Plaintiff's demand, which is why this issue rarely arises). If a defense attorney makes an offer of judgment, and then the Plaintiff doesn't accept the offer, they need to get a jury verdict in an amount greater than the offer, or the costs are flipped. So, typically, if there is no OOJ and the Plaintiff wins even a nominal judgment (it can be a dollar) the Defendant always has to pay their costs, which can be substantial. When a Rule 68 offer is made, it's a carefully calculated amount that the defendant thinks the plaintiff can't get in a jury verdict, even if they win, but it's typically less than the demand.
The reason a Rule 68 offer is almost never "full compensation" is that a Plaintiff's demand for settlement will typically be somewhere in the area of 3x the amount the Plaintiff's attorney estimates the case to be worth. The Plaintiff is informed of this by their lawyer, so they don't have unreasonable expectations. If you think about it, this makes sense from an ability to negotiate perspective, with the logic being that the Plaintiff wants to get as close to full value as they can, and the defense needs their client to think they've saved them from some huge judgement. If a plaintiff demanded only what the case was worth, it would have no chance of settling for true value, or if the case were to settle, the Defense lawyer wouldn't be able to move the Plaintiff down off their number in any substantial way. This way, the lawyers can play their game negotiating the case down to a fair value. Plaintiff gets what their case is worth and Defendant feels like their lawyer saved them from catastrophe. It's all illusion.
The First Circuit recently joined the Second, Fifth, Seventh, Ninth, and Eleventh Circuits in holding that a Rule 68 offer made prior to class certification and rejected by Plaintiff does not moot the Plaintiff’s claim. The Plaintiff, a private high school, brought the action against the corporate developer of a college-entrance exam, alleging violations of the Telephone Consumer Protection Act and an analogous state statute related to unsolicited faxes it received. Prior to Plaintiff’s deadline to move for class certification, the Defendant made an Offer Of Judgment, offering Plaintiff the amount it could receive under the two statutes for each fax. Plaintiff did not respond within 14 days, rendering the offer withdrawn under Rule 68, and instead moved for class certification. Defendant then moved to dismiss, arguing that the withdrawn offer rendered Plaintiff’s claims moot and divested the court of subject matter jurisdiction. The district court denied the motion, holding that Plaintiff’s claim was not moot, but certified the question of whether an unaccepted Rule 68 offer, made before certification, moots the entire action and deprives the court of jurisdiction.
Generally speaking, aside from very specific types of cases involving contracts, or specific types of statutory relief, a Plaintiff typically includes counts for things like NIED (negligent infliction of emotional distress), pain and suffering, loss of consortium, loss of future earning capacity – these are a few of the types of counts whereby there is no specific value a defendant could ever point to being "fully satisfied" – the reason being, a jury needs to determine the legitimate value of these claims unless the Plaintiff accepts a settlement award whereby he/she/it feels as if it's fully satisfied.