As far as I know, It all depends on the installer and what tools you have available. With option 1, you're going to need a Punch Down tool, and with option 2 (which you already said you had) you need a crimping tool.
For example when it comes to cat 6. I've talked to installers that simply can't stand crimping cat 6 with RJ45 connectors because it just takes way too long during big jobs. They use option 1 even though it essentially costs more per connector. They can justify it with time saved from work hours over the long run.
If you're asking which method is better. Test it for yourself. Run the Cat 6 cable and connect the cable to option 1 & 2 and test a file transfer. I'm pretty sure it's going to be the same thing as long as you didn't make a mistake during crimping. And make sure you keep the twists of each pair as close as possible to the edge of the connection.
Is there an advantage to using a wall jack and doing my own wiring?
I'm going to assume, when you say "my own wiring", you're referring to using a crimper with RJ45 connectors.
If that's the case, You can use a professional looking wall plate in either situation.
Option 1
Option 2
As you can see, either way works.