I haven't found the one I like and I created it myself, Qbserve.

It fits most of your requirements:

• native stand-alone OS X app
• detects idle time
• comes for a flat fee ($30, though it will be $40 in future)
• no separation by client but you can create specific categories for your projects and assign different windows to these categories
• no billing support yet, but I will probably add some kind of data export later

It also automatically recognizes productivity for sites, apps and chats (you can assign Skype, Slack and Telegram chats to different categories).

Plus all the tracked information is stored locally on your machine in a SQLite database.

I haven't found the one I like and I created it myself, Qbserve.

It fits most of your requirements:

• native stand-alone macOS app
• detects idle time
• comes for a flat fee ($40)
• supports projects and tracks time for them automatically (based on opened documents and web pages) but it's also possible to assign time manually
• generates invoices from the tracked time

It also automatically recognizes productivity for sites, apps, and chats (you can assign Skype chats and Slack teams to different categories).

Plus all the tracked information is stored locally on your machine in an SQLite database.