Is there an auto log-in function for Time Doctor?

No.

Passively logging hours like Rescutime does not result in the level of accuracy we strive for at Time Doctor. So if you want a general idea of what people are doing RescueTime is OKAY. If you want to know exactly how many hours a person works, exactly when they start and finish work, exactly what they were working on, then Time Doctor is a much better option.

For example, if the person is searching on Google, is this related to work or just something related to their personal life? Rescue time cannot know this properly.

To know how much work a person is doing they have to log their time actively. Then we combine this with multiple other features to confirm they are working effectively such as:

1) Screenshots

2) The user is put on a break if they are not active on their computer for three minutes
(this idle time can be changed, all the way up to one hour).

3) Recording which websites were visited, which applications were used, and tracking how long each was used.

So for example RescueTime could show the person is working 8 hours per day. But really they were only working 5 hours for that day and the other 3 hours were personal activities.
With Time Doctor you get a totally accurate picture that if it says 8 hours of work, it really is 8 hours (not just 5, 6, or 7 hours).

So yes it's easier for staff when they don't have to do anything, but there is a big downside:
The time recorded will not be nearly as accurate and you miss out on a lot of the productivity advantages.

Another reason is that with Time Doctor, because you are always aware of what you are working on now, it makes you much more aware of when you are wasting time. So, it helps individuals to know in real time
(not at the end of the week) how much time they are wasting and to stop wasting that time.
It's very useful to have a constant reminder of what you are working on how and how long it is taking you.