Your idea doesn't sound cheesy at all, it sounds cool and original. :-)
Quick answer: yes it can.
Long answer: What you're describing is cool, but in a technical sense you're so far just wanting words, links and graphics. Twine can do all of those things for you, but so can dedicated website makers like Wix etc. (By the way, I've never used Wix, I'm not on commission for them or anything.)
If you're going to learn either Twine or Wix/Wordpress/Weebly to make websites with, I would recommend Wix/Wordpress/Weebly because they can do things Twine can't do, like simply let you set up pages with lovely typography and allow you to drag and drop pictures around.
With Twine, you can set up lovely typography and place images exactly, but it's not so straightforward as Wix etc.
The benefit Twine offers is that you can easily create code for games – you'll notice most of the other questions here are about how to make health systems, and inventory systems, and clocks that simulate time passing, etc. These are things that Wix et al just don't do, so we use Twine (I think generally because it's simpler to learn than hardcore development languages like JavaScript).
So if you don't need to make games, I would advise you that, although Twine can produce normal websites, making a normal website with Twine is a bit of an inefficient way to do it. Wix or something would be better.
If you have an interest in making games/interactive fiction, though, Twine is awesome.
I hope my reply doesn't come across as trying to discourage you from learning and using Twine – it's really great, and if you pick it up I and the others here will be delighted to help you out. My concern is that it seems a little like you're asking if a sword can open a wine bottle. The answer's yes, but it actually sounds like you need a corkscrew instead.