Connections gives you 16 items to fit into 4 groups. You get a limited number of tries. If your group of 4 is correct it drops out, otherwise you lose a try. You can’t move them around, except randomly.
This is the easiest approach I have figured out for using Obsidian as a support tool for the puzzle.
Use the Kanban plugin. Create a box. Put all the NYT puzzle items into it. Create 3 more boxes. Move items around (look in the menu tree) until you have 4 groups of 4 that make sense (or 3 that do, and you have no clue what joins the last group).
Play the NYT puzzle, using the groupings you made.