Well, partial success again. On the up side, Z-axis assembled from the linear rail and bearing sliding elements does work; there is no play and the motion is nice and smooth - see the video.
On the down side it is still not pulling out the drill bit from the mat completely. I have two possible explanations - since it's a belt drive, Z-axis with mounted Dremel flex shaft slides down from vibration at the short moment the steppers are disabled; or the motor simply doesn't have enough juice to lift the axis and pull the drill bit out of the mat at the same time (there is apparently some resistance).

One of the expensive materials used in Contraptor set is Teflon angle. I had tried to find it cheaper than on Mcmaster.com, unsuccessfully - it is only available in one other place online, and it's even more expensive there, around $30/ft. Ultra High Molecular Weight Polyethylene (UHMWPE), which has coefficient of friction fairly close to that of PTFE, might be an inexpensive replacement. I ordered some UHMWPE to test out and built a couple of sliding elements. Turns out, the performance is as good if not better than with Teflon.
There are couple of things that are different in making sliding elements with it though:
Thinking about ways to increase the feedrate after disappointing tests of the minirouter, I came back to the design of linear bearings. With existing components, feedrate increase can be achieved either via multistart leadscrews or the belt drive. Multistart leadscrews are a) expensive (Contraptor needs 2 times the axis length), and b) require new/custom collars to secure axially. Concerns with the belt drive are: a) is there going to be enough torque to move around heavy gantry sliding on Teflon bearing pads? and b) if there is, will the belts stretch, introducing inaccuracies when accelerating?
Obviously, a linear rail with ball bearings would alleviate these problems: a belt drive would have less friction to work against, and if it wasn't strong (or accurate) enough, then just one multistart leadscrew (instead of two) could be used to drive a stage with reasonably fast feedrate.
This weekend I tested the linear drive with ACME leadscrew positioned in the center of sliding element profile. In summary, with good alignment of shaft mounts, it works extremely smooth without any visibly detectable backlash. When shaft mounts are not well aligned, the torque needed for rotation gets uneven unless there is a small axial slack.
Over the last couple of weeks I tested the linear motion design where leadscrew is located in the center of the sliding element profile. This allows the room for leadnuts to move and be precisely aligned parallel to the rail, which eliminates entire host of problems like whipping, high friction, uneven torque that I had when the leadnuts were essentially fixed to the corner of the sliding elements once and for all. Because leadscrews moved, I also had to modify the shaft/leadscrew mounts, the motor mount and the belt holder thingy. So far I tested center shaft arrangement with 1/4" all-thread leadscrew and the belt drive. Both appear to work well. I have not yet tested ACME leadscrew with delrin nut, the arrangement with belt-connected dual leadscrews and shaft-connected dual belts.
Over the last couple of weeks I made some progress on the following things:
Adhesive printed labels as templates - it's kind of obvious idea I hadn't thought of before for some reason. Easy, fast, and very convenient when making different designs of something - such as the sliding elements. The accuracy is much better comparing to marking holes manually. With the labels, I quickly made several components for the construction set - mounts for the threaded rod, mounts for steppers, and several designs of the linear sliding element.
While I still need to test a few things such as vacuum paper pickup/placement tool and a vacuum table made out of cutting mat, I've been thinking about the coordinate robot that would drive the rotary cutter. I had started drawing sketches of the frame and the robot, which would be built from MDF sheets, shafts, pulleys, belts, etc, but then realized that it would take a lot of work to actually build something like that and then, if something doesn't work as expected, to change the design and re-build it, partially or entirely, again. What I need at this point is a quick rig to test the cutting of straight lines, circles and curves, precision of paper pickup and placement - with minimum time investment. Something like CNC machine but a bit more flexible than that.