The red trails behave exactly as you say when I turn down the
brightness, I guess that's a bad sign. It seems weird that low red
emission creates more red on the screen, but I don't pretend to
understand how the tube works.

The output transistor for the red colour is driven into saturation
because of low feedback, thereby causing the bleeding effect. This
problem is most likely caused by a low +200V to the CRT board, or by a
weak CRT. Replacing the electrolytics on the +200V line is normally the
first thing to try, turning up the screen voltage (G2) is a good measure
if the CRT turns out to be a bit weak.

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Met vriendelijke groet,

Maarten Bakker.