How to sell my current solution to my advisor?
In general trying to sell things to your advisor puts you on dangerous ground.
For example, the way you pitched your research in the question makes it seem like there are no problems that bar publication with your approach (as pointed out in the comments, fast approximations are generally publishable if they are novel). So I can't say why your advisor is hesitating. From the information given, it seems your advisor 1) is hesitating for no reason or 2) has a legitimate problem you didn't describe. If it's the 2nd case, be worried. Someone who brushes over and can't articulate legitimate problems with their approach is not someone who usually puts out high quality research. A big part of getting a PhD is learning how to be critical of your own work, and your advisor can't help you learn this if you don't listen to criticism.
Rather than sell to your advisor, I would open up more communication with them. Find out more about their reservations. Really try to get to the root of why the advisor doesn't want to publish. If your research has merit, that will come through in an honest investigation of its merits and its flaws. If your research has flaws, then listen to people who point them out, understand the flaws, and work on figuring out how to fix them.