That tactic has been shot down by the courts, but there are some big loopholes to it. Any acknowledgement of the debt (normally in writing) or any token payment by the debtor can reset the clock. Sometimes, debt collectors employ shady practices to get the statute reset, such as telling the debtor, "If you pay us $10, we'll stop calling you for three months." This token payment will reset the clock.
Once upon a time (40+ years ago?) accepting partial payment for a debt using an instrument that declared something to the effect of, "for the satisfaction of all debt", was enforceable. This became a problem when automated payment processing systems became prevalent. Savvy debtors--presumably mostly lawyers--could write a check for $10 that extinguished a $10,000 medical bill once cashed. Needless to say, the law was quickly changed before it became too prevalent--much more quickly than laws designed to protect consumers.
Unlike the loophole regarding debt acknowledgment you mention, this loophole was an artifact of mercantile law. Terms on negotiable instruments are typically strictly enforced as compared to regular contracts, to ensure maximum efficiency of payment systems. (Or at least the rules for what's enforceable are very brightline, and aren't particularly concerned with fairness. Magical phrases are still very much important.)
"The takeaway from these two cases is that accord and satisfaction can only settle a disputed debt when both parties have knowledge of the outstanding issues to which the debt pertains, and the party negotiating the check has reasonable notice that by depositing the check the dispute is completely settled for the amount of the check."
I wonder if you could put a 'warning sleeve' or something around a check that is sent to someone who may process it automatically.
