Is there a way to automate time tracking as in the real work value can be generated by the time difference of when the ticket wasn't in a handled status to when it was in a handled status? And then, if the ticket is put on hold and out of handled status then it isn't racking up time.

No we don't have such.
And IMO this is not a good way to calculate spent time, sure "on hold status" may not be used as required, and you'll soon have much more work hours that hours in a day 😉
But it can be a good way to calculate efficiency, when "on hold" is a request for completion.

We use it for production tracking and it would be very beneficial for the amount of tickets if our production guys didn't have to manually start and stop a timer or enter time spent on each part. Rather, the time would be tracked while it is in a handled status.

This seems a too much simplistic approach...
What about lunch time ?
What about a ticket started in the evening and ended the day after or possibly several days after, including off days or hollidays...

Our time is tracked per hour spent on project, not per work day - we are ETO.

And, we have production associates who can update status on each ticket so inaccurate hours wouldn't be an issue.

It's just going to be a hassle to have our guys log hours on each different part that comes through their workstation because of how many there are.

Our time is tracked per hour spent on project, not per work day - we are ETO.

You can record hours, just change global parameter.

And, we have production associates who can update status on each ticket so inaccurate hours wouldn't be an issue.

Sure it will not be so easy.
Same responsible can have several tickets on status "in progress" for same period.
I start working on ticket, set it in progress. Then receive urgent ticket, so open new one, to work on it and set it in progres : face to urgency, sure first ticket is still "in progress", so time recorded is doubled.
Also if ticket is opened (set to "in progress") by one user and closed (set "done") by different users, who is concidered to have done the job.
As I explained, it is not impossible, it is just "much more complex" than a simple difference between dates...