For me, the biggest flaw is the hipster image it has sometimes, mainly created from the ******** comment wars on the internet or the new breed of high artists, playing 20 min melodic minor solo basking in their own musical dawn.
As far as it is true that jazz requires some longer listening to get used to it, I find it very unproductive that many fans present it that way right away, because actually there are artists that sound accessible, let's call it that way, without any(and I am not really speaking about the smooth stuff at the moment). I found that by simply enjoying the music or having a passion for it and presenting it as a possible alternative to others positively I got a lot of my peers listening to at least SOME jazz, be it Joao Gilberto, Oscar Peterson, Miles, Sonny Rollins, some even found themselves to be interested in Keith Jarrett. Some do more often, some not so much. But my point is they do, they like it and they present jazz to others so they pass the good message.
I think what really is enough, speaking of the open mind, to have an open mind. For other genres. There is a big difference between 'What is this crap' and 'I don't really like this'.
The only thing for me that makes jazz less popular is speaking of it like that. Of course it will not surpass pop audience, but on the other side that never was the point I think. Nonetheless, I rather speak of it as a great genre where almost everybody can find something that he/she will like if he looks for it. And presented this way, without insulting that person's taste or views, sending some stuff in, people I did this with almost always found something. And this is that kind of attitude and situation that applies to a lot of things happening in other aspects of life as well.
But, that is just my view on this.