Distortion-free map of all continents
I've been searching for a map that shows all continents in their actual size that is free of projection distortion, to no avail


That's because such a thing does not exist. A projection by definition has distortion, otherwise it would not be a projection.

The only way to have a distortion free view of the planet is by having a physical globe, and looking at it with both eyes so your brain gets the three dimensional perception (assuming going to space is not an option). Anything else will result in distortion.

Now that we have this covered, there are all kinds of projections with all kinds of distortions. Distortions can affect areas, distances, shapes, etc. to various degrees. It's your choice to determine which types of distortions and their magnitudes are acceptable to you, personally. There is no need to reinvent the wheel, there are enough projections out there already.