The higher the variety of notes (out of the overall 12 sounds in an octave) in the song, the less this becomes possible.

If your song is really simple, e.g. only consists of the 3 notes that make up a major triad (root, third, fifth), then this is definitely possible and you can just use natural thirds and natural fifths.

But as you start adding more notes, more chords and perhaps change of keys etc, it starts to break down.

That's the reason why J. S. Bach wrote The Well-Tempered Clavier. It's a collection of 24 preludes and fugues, in each possible major and minor key.

The basic idea was that if every prelude and fugue sounded good on an instrument (organ, harpsichord etc.), than it meant that the instrument was "well-tempered".

Using natural tuning instead of 12-TET would have resulted in some pieces sounding very good and other sounding very bad.