Would using a leadscrew with 1 or 2 mm lead, en lieu of 8 mm, result in a better printer?
From Accuracy vs Precision and Threaded Rod vs Leadscrews in 3D Printers. I have highlighted the relevant parts:

In general, FFF/FDM printers use relatively infrequent, small, precise
movements on the z-axis and consistent, fast movements on the x and y
axes. A single start leadscrew with the tightest pitch possible
(highest thread density, smallest pitch) is generally going to be your
best bet for the z-axis, while you may or may not need something a
little steeper to get the speeds you'd like from your x and y axes.
While this may seem somewhat arbitrary given the precision of movement
you can get from a stepper motor, an important factor to remember here
is torque.
A more aggressive leadscrew will require more torque to drive. We have
one kit printer we bought a couple years ago that has an overly
aggressive multi-start leadscrew for the z-axis. The small motors
included in the kit do not have the torque required to reliably start
upward movement of the carriage, leaving it sitting there skipping
steps until the carriage is given a little upward nudge to get it
going (no, it's not a lubrication issue or a driver that needs turning
up).

So, a 2 mm lead is preferable to a 8 mm lead, as not only is less torque required, but also more precise movements are obtainable.

With respect to the layer heights, I found this nugget of information, from Ditch the threaded rod in your RepRap 3D printer and upgrade to a lead screw z-axis

It's always better to use layer heights that are a multiple of your
full step. If you trust in your micro-stepping you will get poor
results because torque is very poor and the motor won't stop very
precisely. Best approach is to use a lead screw (whose longer step
allows the gravity to work for you and eliminate backlash) and a
stepper driver configured to 1/4 micro-step for low noise, but not for
micro-layering. I always use multiples of my full step and have
printed 0.040 (1 step), 0.080 (2 steps), 0.120 (3 steps and so on), 0.200,
0.600 and 0.800 mm (of course using 2 different nozzles). Do you need
more than that?

See also Reddit - Lead screw opinions

the general consensus is that a 2 mm lead is preferable to a 8 mm lead:



Lower the lead the better for an Z-Axis screw to allow for more precision.
No need for anti-backlash nuts for a Z-Axis if it has a decent amount of mass/weight to it.



1 mm will slow the printer down, but provide 0.005 mm (5μm) vertical movement:


That's going to be awfully slow when homing. While you really don't
need speed for the z-axis, there's not much point in going with such a
low pitch... you won't be printing anywhere near the layer heights
that can achieve. In the end I guess it's not a big deal, but I don't
see any reason to go below about a 4 mm pitch. A 1 mm pitch with a 1.8°
stepper gives you 0.005 mm increments. Anything under 0.02 mm (which is
sort of ridiculous anyways) is just unnecessary. Just my thoughts.


(As Tom points out) An 8 mm lead can result in the weight of the X-axis gantry (especially in a P3Steel) overcoming the idle torque of the stepper. As a result of this, the X-axis gantry can end up sliding down the leadscrew, in particular at power down:


I have lead screws on my Z and it readily falls down if you kill the power. Regular threaded rod is much better at keeping it in place without power.

and

That's one of the problems with 8mm lead ;) With 2mm lead it will hold itself. The stepper motor provides a fair amount of resistance even when powered down.


TL;DR
Be wary of leadscrews with a lead, as opposed to a pitch, of 8 mm, as there are disadvantages, when compared to a lesser lead of, say, 4 mm, or 2 mm (or 1 mm for leadscrew with a pitch of 1 mm):

Higher torque may be required
Less accuracy
Possibility of slippage, of the X-axis, once power is removed, or stepper is disabled1
Anti backlash nuts required

However, their major (if you can call it that) advantage is a higher maximum movement speed.