Getting past falsifying of timesheets is _very_ difficult. It's the norm rather than the exception, at least in my experience.
When I used to sling burgers, etc, my employers would regularly retroactively clock me out for breaks I wasn't allowed to take. We were clocked out when our shifts were supposed to have ended, regardless of when we were allowed to leave.
It was normal to work 10-12 hours and get paid for 7 (at $4.75 & $5.15, none-the-less). If you report it, you're fired and blacklisted.
While I was under 18, I would be clocked out at 11pm on any school weeknight (state law). The catch was that I was on closing shift, so I couldn't leave (locked doors) until we were through -- usually ~1am. 2 hours per-night unpaid, plus a 30 minute deduction from my pay for a break I was never allowed to take.
All of that is illegal under a variety of state and federal laws. However, enforcement of those laws is either non-existent or is a tiny slap on the wrist ($500 fine for stealing an aggregate of ~$10,000 in wages over a year in the case of one place I worked at that was finally caught).
I'm getting on a tangent, but my main point is unpaid extra work is rampant among hourly employees as well as salaried.
I completely agree with your premise, but I think we'd see the same problems come up in a different form. Fixing that problem requires putting at least some bargaining power back in the hands of employees, and that's difficult in most places.