I believe the extended warranties are administered by an outside company and are contractual obligations that would be fulfilled, though you might have little recourse (going back to Sears to complain) should you not be satisfied.
Sears has dug their own grave when it comes to retailing in the 21st Century. While their brands like Kenmore, Craftsman and their automotive brands were top sellers, much of this business model depended not only on good products at fair pricing points but also was heavily dependent on availability of consumer credit. Being a powerhouse of retailing, they had enormous resources to provide worthy customers an open credit line when others, independents, etc. did not have access to such marketing tools. At 18% interest, the credit department became the cash cow of the retail model.
Sears was one-stop shopping for nearly everything including home improvement. New siding and gutters as well as back-to-school clothes for the kids and a Kenmore washer to keep them looking smart, all in one stop. Short of cash and your refrigerator conks out? Head over to Sears and bring one home today, nothing down. Satisfaction guaranteed or your money back. You can imagine the quarterly reports for Toughskins jeans sales being rather insignificant compared with the line item for interest income on revolving credit balances.
Once this credit model of retailing began to erode, Sears didn't change with the times. Coming out with their Discover Card in the 80's was too little, too late. While it was a good product with attractive terms, it couldn't hold a candle to the name recognition and worldwide acceptance of Visa and Mastercard.
In the mid 70's, First National Bank of Omaha challenged and won the Federal law that prohibited the sale of financial products (credit cards) across state lines. Previously, all banks offering Bankamericard and Master Charge products were regulated to in-state institutions. After the Supremes ruled against this, nationwide sales of credit cards became the norm and credit-worthy individuals across the country saw their mailboxes fill up with competitive offers for lower interest rates and generous credit lines on credit cards from banks they'd never heard of. The availability of competitive credit products drove up consumer spending numbers to all-time highs within a decade. Other than the slight bump from Discover card accounts, Sears missed much of this bandwagon relying on their established base of credit customers as well as dwindling numbers of new customers to the Sears family. Their refusal until the early 1990's to accept Visa and Mastercard for payment also hurt sales numbers dramatically. It wasn't an instant plummet to the bottom, but a steady decline.
Relying on their aged, tried and true business models, Sears didn't keep up with Lowes, Home Depot and a myriad of automotive sales and repair outlets that exploded across the country. Then the Wal-Marts, Targets and the like who also offered a similar, but more basic-need range of shopping under one roof; a set of new tires, clothes and ice cream sandwiches - all on the Visa card, tied to the untapped pool of equity in your home.
I think the writing was on the wall when Sears merged with K-Mart, never a smart powerhouse of retailing in itself. We've lost so many of our big names over the years, but Sears has lasted longer than expected.