by leaving agents logged in your are severely inhibiting management of these agents for a few reasons but mainly due to reporting.
The idea to have them log in is that they are ready and able to take calls
If they have issues with agents not being logged in when calls arrive, they have to manage with the person not the technology in my opinion.
As someone suggested, they could play around with the CTI toolkit and/or web services sdk (depending on their software version) to make this "automated" in some shape or form.
It doesn't make sense to automate it though...