At What Rate Do Ions Leak Out of a Plasma Membrane Segment That Has No Ion Channels?
I'm going to mostly skip over the title of your question and focus on the confusion in the body, because the question in the title is a bit of an XY problem. It doesn't make much biological sense to think about a membrane that is truly that resistive (that is, I am unaware of any example neuron, or even any other cell, that has that property).



If there were truly no channels of any sort, you could indeed consider the resistance to be very high, at least as high as one could measure in patch clamp. Additionally, what matters is not the resistance itself but the ratio of the transverse to axial resistance; relative to the axial resistance the transverse resistance is already very high.


  From what I understand, one of the primary "benefits" of myelin is that it aids in increasing the transverse resistance of an axon


Your understanding is not completely correct. The insulating benefit of myelin (ignoring other organizational benefits) is not only related to the transverse resistance, but importantly to a decrease of the capacitance of the membrane (see Moore et al 1978 or Richardson et al 2000 for some example simulations; also any reasonable neuroscience textbook will explain this). You can test this yourself in a simulation environment like NEURON.

Myelin decreases capacitance by effectively increasing the distance between the 'plates' of the membrane capacitor (see this answer). 

In order to charge a long length of membrane with channels on only at short segments, you would need enough charges to come in within that short segment to charge the entire length of axon, and the potential would decay with distance. 

Myelin does also affect resistance, but that's mostly important because the membrane does not consist only of a phospholipid bilayer and there is non-zero leak. For example, there are channels under the lipid bilayer that establish a resting potential (Chiu &amp; Ritchie, 1984).



Chiu, S. Y., &amp; Ritchie, J. M. (1984). On the physiological role of internodal potassium channels and the security of conduction in myelinated nerve fibres. Proceedings of the Royal society of London. Series B. Biological sciences, 220(1221), 415-422.

Moore, J. W., Joyner, R. W., Brill, M. H., Waxman, S. D., &amp; Najar-Joa, M. (1978). Simulations of conduction in uniform myelinated fibers. Relative sensitivity to changes in nodal and internodal parameters. Biophysical journal, 21(2), 147-160.

Richardson, A. G., McIntyre, C. C., &amp; Grill, W. M. (2000). Modelling the effects of electric fields on nerve fibres: influence of the myelin sheath. Medical and Biological Engineering and Computing, 38(4), 438-446.