Why would I want a mechanical/automatic watch when quartz is more accurate?

Simple. Quartz is clearly better on accuracy. But there are many other advantages and pleasures from wristwatch ownership beyond just measures of precision levels that are beyond the notice of many people.

Frankly, quartz watches and many other technologies don't really do anything significant to better people's lives. People with quartz watches are no more reliably on time than people with mechanical ones. People driving cars with manual or automatic transmissions still get where they are going equally well. People still enjoy music about as much as they used to, even though CDs play it more clearly that tape or LPs did. You are not likely to have any smarter thoughts simply because you wrote them down with a computer than with an ink pen. You can easily spend as much time playing golf or football on an accurate computer simulation game as on a real playing field, but the experience is not any more fulfilling at the end of the day. And you can certainly do a lap around the lake faster in a speedboat than in a rowboat, but what have you really accomplished?

The newer technologies often gain a level of efficiency that makes them uninteresting. In many cases, the older ways and technologies were more than sufficient, and it is their minor failings that give variety and character to doing things that way. With the older ways, you usually have to be more aware of details, understand more of what you are doing, and take more time being involved in the process. That greater interaction makes the process more personal and enjoyable for some people.

With the newer ways, you can be pretty assured your quartz watch is on the right time, your car's automatic transmission won't miss downshift on the way home, your CD will play exactly the same as it did yesterday, your computer will catch and correct your typos and misspellings, your video game won't stop in the middle because of rain or a player injured in a tackle, and you certainly won't be bothered seeing much of the detail and wildlife on the lake at high speed from your motorboat. How boring.

Mechanical watch enthusiasts often compare the movements, the finishing, the level of adjustment, types of certifications, performance under different circumstances and other esoteric measures of mechanical timepieces.

Quartz watch enthusiasts compare mostly accuracy measures.
So if efficiency is your main desire, then quartz is for you. If you are tired of efficiency and want something interesting instead, try a mechanical watch.