The only use of e-sports for a video game publisher (unless they have invested in MLG, or another e-sports network) is for advertizing purposes, to sell more of the game. Nothing more, nothing less. So I don't think Nintendo should focus on making F-Zero an e-sport. I think they should (if they want to make an F-Zero at all) focus on making it as good as possible, and if it's sensible for marketing purposes to try and make it into an e-sport, sure, why not.

TL;DR: No game should be created with the goal of making an e-sport. Unless MLG starts developing their own games.

I totally agree..

but trying to compete against themselves to make both popular won't work out well. Mario Kart is just more popular, no matter how you look at it so it won't be easy

If people wanted an e-Sport racing game, there are a lot of other games that could take it's role too, so it would be extremely difficult