Solved it. Needed to add it to the .AddPlugin("... table") method. Every example online showed it as adding it as .Toolbar("... table").
You need to .AddPlugin() to make the plugin available. Then you can add it to the .Toolbar() or add it to the menu with .RawSettings().
Refer to https://www.tinymce.com/docs/advanced/editor-control-identifiers/#toolbarcontrols to see what buttons/functions require what plugins. Core is included by default.
Refer to https://world.episerver.com/documentation/developer-guides/CMS/add-ons/customizing-the-tinymce-editor-v2/ on configuration in Episerver.
I've been trying to add the table toolbar to my TinyMCE and not having any luck. Just checked Chrome Dev Tools for a possible error and noticed the "OTS parsing error: name: faled to parse table" message. I assume this is related to TinyMCE. I'm trying to recall how to bring up the modules loaded for TinyMCE using Chrome Dev Tools console, but can't recall.