Back around 2010 I worked for a company which built software to catalog residential distribution utility poles and the lines emanating from the poles to the houses.

To elaborate on a few of the statements from the post

    Poles in my neighborhood tend to have consecutive numbers. I don't think this was carefully planned.

The sequential ordering of pole numbers is definitely planned, and fits in with a scheme developed back in the 60s-70s which made documenting the poles lifecycle in paper records much easier. The poles are numbered sequentially with prefix numbers generally indicating which substation district the pole belongs within. Distribution utilities think about their infrastructure as a network of substation circuits. You should be able to tell which circuit you are on based on the numbering on the pole.

    There are three because that is one way to change the three-phase power to single-phase, something I wish I understood better. Truly, we live in an age of marvels.

Not all distribution systems use three phases, some use two, some use single. It all depends on where you are in the system and the general layout of the circuits within the distribution network. Also not all distribution is 7kV, some are 12kV, and some are 25kV. Again this all changes based on the properties of the system, and the qualities that are being optimized for by the power engineers.

     I have tried to imagine what the number-burning device looks like, but I'm not at all sure. Is it like a heated printing press, or perhaps a sort of configurable branding iron?

This can be done a number of different ways, but one of the common ways is with a laser engraver. This is a fairly recent practice, so I am not as versed in the operational details as with traditional tagging. Another means of tagging which is not covered in the article is carving the numbers into the pole with a physical etcher, similar to a wood router.

    Without the Internet I would just have to wonder what these were and what OSMOSE meant.

OSMOSE was at the time of my work in this field, the largest such company that dealt in the support and maintenance of residential utility poles. They have some very interesting proprietary mechanisms for managing inventories, and are generally a very smart group of people. My guess is that they collect all the inventory and survey maintenance of poles using mobile LiDAR and machine vision now, as that was what we were speaking to them about at the time.