The answer has to do with enforcement.
Under the new Singapore Convention on Mediation, International Commercial Mediated Settlement Agreements (ICMSA) are given a new elevated legal status.
In signatory countries, a party is able to receive relief from a court (or other competent authority) by providing direct evidence that their agreement resulted from mediation. Thus, the Singapore Convention gives mediated settlement agreements the same legal status for enforcement as an arbitration award or court judgment.
This enforcement benefit is not available for settlement agreements negotiated directly by parties or their lawyers outside of mediation.
Consequently, the enforcement advantage is one extra reason why parties and lawyers should consider involving a mediator in settlement negotiations.