In my opinion it's HUGELY important.
My thoughts; sexy =
1.) Free press, lots of it, results in free traffic and traction SO increased chance of success. As an investor, getting behind an 'animal' capable of producing either viral growth or free press (helping reach users) delivers a product with free marketing built in and that is a huge overhead ticked off with growth potential beyond most ideas.
2.) Investment is easier to raise, you get more of it, faster, at a better deal and from better investors SO increased chance of success.
3.) 'Sexy' tends to attract A Class employee's, who in turn attract and build an A Class company with the potential to execute the idea or pivot SO increased chance of success.
4.) Directly related to No. 3 is the increased retention of staff. Again, hiring, knowledge and skill loss, downtime, etc can kill a startup so this is a huge advantage.
5.) Company moral - It's a lot more fun to work long hours on an idea that could change the world. It is more rewarding and so going the extra mile, maybe even for a lower wage, can result. Maintaining company focus and moral is vital to success.
6.) Finally, Users. If it's sexy it's sticky. If it's 'viral' it can grow fast, become addictive and present users a product they just love.
Sex sells. Really, a better question is what defines 'sexy'. It's the job of the founder(s) (entrepreneur) to communicate the vision to the press, investors, potential employee's, current employee's, etc. The skill is in making the idea appear sexy. Many successful companies simply took an existing idea and put a 'sexy' image on it, sold the idea better, resulting in 1-6 and bang, you got a winner.
As for being new, innovative, it's obvious that's important. And as for Crazy, well, what's sane about any startup?