Don't get me wrong; I'm not advocating for time sheets. I'm advocating against overtime exemption.
Even so, all other things being equal, if the tedium of timesheets is on one side of the equation and all of the exploitation of unpaid labor is on the other, I'd still rather not be exploited. Working for free, which is what unpaid overtime is, is unsupportable.
There are many examples of non-exempt professionals who deal with this without resorting to spyware or coercion. IT support specialists, paralegals, and lab technicians all have systems that work: simple start/stop time logs or weekly attestations, plus manager pre-approval for overtime. No one is tracking keystrokes and no one is forced into surveillance. It's about accountability. You attest to your hours, managers approve exceptions, and overtime gets paid. That's the balance.