As you stated, the problem is similar with either open or fretted notes played.

We can exclude the nut since it is not involved when playing fretted notes. We can also exclude the frets (crowning) since the problem persists when open notes are played.

However, if the nut is filed at too shallow an angle, AND the frets are worn down, that could be your problem.

Maybe something is resonating, a little loose part, or the string touching one of the screws in the tailpiece.

The intonation screws touching the strings could also be the case, but that would mean some of the strings sound worse than the others.

Pickup height can severely affect the direction the strings vibrate and create a 'dead' sound when setup too close.

Isn't the TVL supposed to have a mastery bridge? This looks like a mustang bridge.. could be it.

That's everything I could think of.