Should I be using random data with Capybara integration tests?

I'm writing a very long integration test for a wizard that has around 15 steps. Each of these steps has around 20 inputs/select boxes.

I started out using static data in my tests, but now I've begun to write stuff like selecting a random value from a select box, and clicking a random radio button for an option. This does seem like it's more capable of catching bugs, for example; one of the buttons on the page might not be rendered correctly and therefore the value never gets saved to the database - this would never have been found using static data that selects the same option every time. Alternatively, I could manually write out every possible option that could be chosen, but that'd take an eternity to do.

I hear that one of the main reasons not to use random data is that you can not explicitly see the data used in your tests and it can make failing tests hard to resolve.

Is this path that I'm going down one to be avoided? or is testing in this manner something that's generally done?