> It doesn't matter how good the intonation at the bridge or nut.

Yes, it does.

> There's the mathematical fact that we cannot get pure thirds and even fifths in modern equal temperament system.

Those are the pennies that don't matter, if your instrument has dollar problems.

If you don't have good intonation, then you can't even properly get the approximations provided by equal temperament.

With good intonation, compensated on both ends, you have a much better experience making tuning adjustments to get better compromises for the music you are playing.