What is a good approach to symbol identification/recognition given a path, instead of raster data
What you are looking for is called "on-line recognition". I have written my Bachelors thesis about this:


  Thoma, Martin. "On-line Recognition of Handwritten Mathematical Symbols." arXiv preprint arXiv:1511.09030 (2015).


And I've created the online service write-math.com as a part of it.

As a short introduction, you might be interested in my presentations.


  Are there any good techniques for doing identification without first rasterizing my paths that may be well suited for my particular instance?


Yes. You can on the one hand make it a fixed-length sequence, on the other hand you can use recurrent neural networks (I didn't try that in my bachelors thesis)


  Is rasterizing to a grid particularly well suited to the problem, and should I just raster and solve this problem more classically?


This works astonishingly well. I achieved better results with the rasterized version (in my master thesis which isn't published yet due to bureaucracy problems at my university) than with fixed-length sequence techniques. But I guess the way to go is to do both.