My app Timing automatically tracks time spent on your Macs. That's perfect for your purposes, as it lets your team keep track of time without having to worry about start/stop timers.
Also, I'm currently working on a brand-new version, launching next month. That version has been built from the start for the needs of freelancers and small agencies, with a ton of new reporting features and optional start/stop timers if you need them. You can sign up for the beta at https://beta.timingapp.com.