The reason for my OP confusion is that it turns out that there are two similarly-named tools with different capabilities. I'll try to explain their differences here.
The plugin tool that I was searching for in my OP is:
Points to Paths (note the plural "Paths")
This tool is available at 2.x, but has not yet been updated to 3.x Here's a screenshot (note the highlighted Line per Vertex checkbox):
3.8.2 (and I assume all 3.x to date) contains a similarly-named (but functionally limited) tool which does not appear to have been available at 2.x:
Points to Path (note the singular "Path")
Here's its screenshot:
To recap, the 2.x tool is Points to Paths, while the 3.x tool is Points to Path.
Note that the Points to Path tool does not contain the Line per Vertex checkbox. This seemingly insignificant difference is crucial. When checked on, Points to Paths will always create a line segment "downstream" from each point. Furthermore, each line segment will contain the unique value from its associated point's Point Order Field, which allows you to subsequently join each line segment to its associated point, which in turn allows you to transfer any other tabular values from point to line.
In contrast, Points to Path will only create a single line for each Group Field. None of the intermediate line segments between points will be created. Additionally, without those intermediate line segments (and their associated id values), no joins back to the point layer are possible.
With its Line per Vertex checkbox off, Points to Paths and Points to Path are equivalent.
In summary, Points to Paths is the more versatile and useful tool. I look forward to it's inclusion at 3.x