By multi-tool, do you mean a rotary tool like a dremel or one of the oscillating multi tool? Personally I haven’t found much use for a use for a rotary tool in my home improvement adventures, but I have friends more into woodworking that love the oscillating multi tools.
In addition to the ones you listed, I would recommend a jig saw, circular saw, and optionally a belt sander. That will cover most small to medium home improvement projects. I found all 3 of those tools for less than $20 each looking around pawn shops. The jig saw is the most versatile, you can cut wood or metal with the right blade, it just doesn’t excel at making straight cuts. That’s where the circular saw comes in.