If I'm logged in to my client's WP site as myself, I can't even activate a service without connecting Publicize to my WP.com account even though JetPack is already connected to my client's WP.com account.
If you are an admin on the site, you can activate and deactivate Jetpack modules without connecting the site to your WordPress.com account.
Jetpack modules, and Jetpack options in general, are linked to your site URL. You could make that global connection with your own account, set up the site and each Jetpack option, and then once you're all done, you could disconnect from WordPress.com in the Jetpack menu, and connect back using your client's WordPress.com account. The options won't be lost, and your personal account won't be linked to your client's site anymore.
If I connect Publicize to my own WP.com account, then activate Facebook with my client's FB login so that her posts go to her fan page, what is the point of having the connection run through my WP.com account? Why should it not run through her WP.com account which is what JetPack is already connected to?
I understand how confusing this can be, and I completely agree with you here: it doesn't make sense to have the connection run through your WordPress.com account, when you do not publish the posts, and when the Facebook account is not yours. In this case, the connection should be made through her WordPress.com account.
I think this makes more sense on a multi-author blog.
When running a multi-author blog, where each author might want to push their own posts to their personal Social Media accounts, you want each author to be able to enable Publicize from their own account. These authors might not want to give access to their personal Social Media accounts to another administrator on the site, so the Publicize connection is linked to their personal WordPress.com account.
Basically, Publicize is a personal service, linked to a personal WordPress account and a personal WordPress.com account. You can choose to share that connection with others if you want to, but others should not be able to enable connections on your behalf.
I hope this helps a bit.