Sears "Power"
Such as it was came on a few fronts.
They had stores nearly everywhere from cost to coast. Then there was their famous catalog which often worked hand in hand with retail locations.
Where Sears really had a killing was with their credit card. One of the main reasons Kenmore appliances were in "more homes than....." was because people bought the things on credit from Sears. Of course there was much more than that; clothing, household furnishing, tools, automotive, etc.... All could go on the card.
Because Sears could move much merchandise they did have "power" I suppose. Rather like the walmart of their time. Because they placed such huge orders with Whirlpool or whoever, they could dictate many conditions.
No, Sears didn't build their own appliances, but they were known for quality and often matched or were better in that area and features than their national branded cousins.
Go back though the years and look at Consumer Reports reviews of say washing machines or dryers. You'll often see that the Kenmore version of this or that Whirlpool or whatever was just as highly rated or maybe slightly better.
Sears didn't just source appliances from manufacturers, but had very strict contractual guidelines and production parameters. IIRC things weren't just taken off a line and a "Kenmore" name slapped on; but a plant would stop and run nothing but this or that Kenmore appliance until an order was filled.