Is there a map that displays every country at its correct relative size?
Yes.
Every equal-area map displays all countries (and other areas) in their correct relative size. Inevitably, they don't show the correct shape (unless you're looking at a globe). Personally, my favourite projections are pseudocylidrical equal-area projections, such as:
Equal Earth projection

Mollweide projection

Eckert IV projection

Tobler hyperelliptical projection

There are others; see this list on Wikipedia.
The pseudocylindrical part means that lines of latitude are horizontal lines. I think that for any non-polar projection in the Earth sciences, this is a highly desirable property. A truly cylindrical projection means a projection onto a rectangle, but that just makes distortions even worse than they need to be.  There are other practical considerations, such as whether the poles are points or lines and whether the outer lines of longitude are ellipsoids or sinusoids, which differ between the various examples I've listed above.