Is it true that in Shakespeare's Twelfth Night, "At the heart of the matter, what the characters truly desire is power, not love"?
This statement is mostly true for Duke Orsino and even Olivia, but not for Viola. Viola's motives for fulfilling love are completely different.
Taking Duke Orsino as an example, we see that he has become obsessed with Olivia. Since his love is an obsession, he actually has a desire to control her, and the less he can control her, the more he becomes obsessed with her. We see both his obsession for her and his desire to control her illustrated in the very first scene. He portrays his obsession by likening his heart to a deer being hunted by his desires, which he likens to hunting dogs. We also learn how his desires for her have also become a desire to control her when he reflects in his final speech in this scene on how if she can love her brother so deeply that she allows herself to mourn for him for seven years, then how much more could she love a man she is in love with. But the way he phrases his reflection is the most telling. He describes her possible act of falling in love as the act of Cupid's arrow driving out all of her other affections, even her affection for her brother, as we see in his lines:
To pay this debt of love but to a brother,
How will she love, when the rich golden shaft [Cupid's arrow]
Hath kill'd the flock of all affections else
That live in her. (I.i.36-49)
Hence, we see from these lines that since he views Olivia's act of falling in love as a cessation of all other affections she feels, we see that his obsession is really a desire to control her affections, which amounts to controlling her, which would also be having power over her.
Viola, on the other hand, is able to separate her feelings of love for Orsino from the duties she must perform with him, proving that her love is much less an obsession. By the fourth scene, we learn that she has fallen in love with him because she sees him to be a good and noble man, just as the captain who rescued her described him in the second scene. Hence, when Orsino asks Viola as Cesario to go and court Olivia on his behalf, thinking that Viola's more feminine looks might be better received by Olivia, we see just how much pain the request is causing Viola, as she states in her aside, "Yet, a barful strife! / Whoe'er I woo, myself would be his wife" (I.iv.43-44). Regardless, she very dutifully fulfills her service to Orsino, even having many conversations with Olivia in which she calls Olivia cruel for rejecting Orsino. Since Viola is very willing to do Orsino's bidding, despite the pain it causes her, we see that she has no desire to change Orsino in the way that Orsino wishes to change Olivia. Since Viola has no desire to change Orsino, rather accept what fate unfolds, we can say that she has no desire for power over Orsino in the same way that Orsino craves power over Olivia. Instead, Viola's love for Orsino is the much purer, truer form of love reflected in dutiful service and friendship rather than in obsession.