My question is how critical is it to completly eliminate backlash...
40 years ago men were sent to the moon. Most of the machines that built the rocket that got them there had backlash to some extent.

The critical bit is reducing the backlash/play/give/flex/strain/vibration/call-it-what-you-like it the whole machine down to workable level and working the machine to reduce the effect of the errors.

If you lock them together then wear becomes your problem. Spring them together with sufficient force to resist the tool pressure and you need monster size motors to turn 'em.
The important thing it to lock them together with the smallest possible clearance. This can mean that, with a poorly formed lead-screw thread, the nut will be slack in places while tight in others (lapping the thread with a nut and abrasive e.g. metal polish can reduce the tight spots).