When I used to bill per hour, I used Harvest - https://www.getharvest.com/ and their mac app https://www.getharvest.com/apps/mac.

The great thing about the app is that if your computer is idle for a while it stops the timer. That was the most annoying thing that used to happen to me. When I went to pick up the kids or something I would forget to stop the timer. But the app automatically stops the timer and asks you to confirm.

I still use harvest to bill my clients using the invoice feature. Their free plan is all I have ever needed.