So, I made a new vault and switched themes and realized in some cases the issue was the theme I was using. I was able to make some simple tables that were padded correctly by Advanced Tables.
However, Advanced Tables didn’t like that I bolded this character, I guess?:

It cannot space the first row correctly, for some reason.
It would be cool if user-friendly table management was built into Obsidian, but I’ll also go ahead and post this separately as an Advanced Tables bug.