Clef change at the beginning of a piece

When changing the clef at the beginning of the first measure, Lilypond just changes the clef for the staff. This is how to keep the staff clef and add an extra clef after the time signature and before the first note.

\version "2.24"

%% http://lsr.di.unimi.it/LSR/Item?id=792
%% see also http://lilypond.org/doc/stable/Documentation/notation/displaying-pitches

% Append markup in the text property to the grob
#(define (append-markup grob old-stencil)
  (ly:stencil-combine-at-edge
    old-stencil X RIGHT (ly:text-interface::print grob)))

trebleToBass = {
  \clef bass
  % Fake staff clef appearance
  \once \override Staff.Clef.glyph-name = #"clefs.G"
  \once \override Staff.Clef.Y-offset = #-1
  % Make sure any key signatures will printed with respect to
  % correct middle c position expected for treble clef
  \once \set Staff.middleCClefPosition = -6
  % Append change clef to the time signature
  \once \override Staff.TimeSignature.text = \markup {
    \hspace #1.2
    \raise #1
    \musicglyph "clefs.F_change"
  }
  \once \override Staff.TimeSignature.stencil = #(lambda (grob)
    (append-markup grob (ly:time-signature::print grob)))
}

bassToTreble = {
  \clef treble
  % Fake staff clef appearance
  \once \override Staff.Clef.glyph-name = #"clefs.F"
  \once \override Staff.Clef.Y-offset = #1
  % Make sure any key signatures will printed with respect to
  % correct middle c position expected for bass clef
  \once \set Staff.middleCClefPosition = 6
  % Append change clef to the time signature
  \once \override Staff.TimeSignature.text = \markup {
    \hspace #1.2
    \lower #1
    \musicglyph "clefs.G_change"
  }
  \once \override Staff.TimeSignature.stencil = #(lambda (grob)
    (append-markup grob (ly:time-signature::print grob)))
}

\relative c {
  \trebleToBass
  \key f \major
  c4 d e f
  % This should not be visible
  \clef bass
  g a b c
  % This should be visible
  \clef treble
  d e f g
}