It most likely would, or at least enough to significantly help. I'd just be concerned with whatever other stuff is in it as well, more than the acetone content itself. When you're painting metal, there really IS no such thing as "too clean." Is it absolutely necessary? Probably not, but it is a cheap way of greatly improving your chances of a clean, well-bonded coating. Depending on the paint, I'd be a bit worried by the oily residue that mineral spirits tend to leave behind. Might not hurt with the right paint, or it might be a killer. That's the reason I mentioned acetone or MEK, they both evaporate clean. I've had particularly good luck with acetone. For any of these, you will want gloves and good ventilation! Don't take risks with your health.
The shrink tube is a darn good idea for a quick fix, no trouble at all there.