Why is Kate the most important character in the play She Stoops to Conquer?

If for no other reason, Kate Hardcastle in Oliver Goldsmith's She Stoops to Conquer is the most important character in the play because she is the titular "She" (who "stoops to conquer").
A young Mr. Marlow has been invited to the Hardcastle home by Mr. Hardcastle, who has chosen Mr. Marlow to be Kate's husband.
When Marlow first meets Kate in act 2, she is fashionably dressed and obviously a woman of class and means. Left on his own with Kate, Marlow is unable to talk with her. His shyness with well-bred women overcomes him, and he stutters and stammers and can barely put a sentence together in conversation with her. He sees a chance to escape her company and simply leaves her standing alone.
She remarks on their meeting:

Was there ever such a sober, sentimental interview? I'm certain he scarce looked in my face the whole time. . . . He has good sense, but then so buried in his fears, that it fatigues one more than ignorance. If I could teach him a little confidence, it would be doing somebody that I know of a piece of service.

In act 3, Kate and Marlow meet again, but this time, Kate is dressed plainly, and Marlow mistakes her for a barmaid he once knew. Kate realizes that Marlow has difficulty talking to real ladies but has no trouble at all talking with women of the lower classes, so she plays along, hoping to get to know him. Marlow become so at ease with her—thinking that she's the barmaid—that he makes a pass at her and tries to kiss her. While Kate struggles to get away from him, her father appears, and Marlow runs away.
In act 4, Marlow is appalled at his behavior, and he resolves to leave the barmaid alone. Kate—who is now pretending to be a poor relation of Mr. Hardcastle—is impressed that Marlow is able to show his true and good character. Eventually, Mr. Hardcastle, too, comes to realize Marlow's true character and agrees to Kate's marriage to Marlow.
Kate's ruse of "stooping" (or pretending to be a member of the lower class) allowed her to "conquer" Marlow and provide the encouragement Marlow needed to win Kate's heart.