The quality of the sound to me seems like an out of tune RD or otherwise skipping cog. I assume you’ve tried micro adjustments with your friction shifter, but if not try to “tune the noise out” with the shifter. If that works you have your answer. If not, maybe look at a stiff master link in new chain or worn freewheel cog. Another issue I find can pop up is with under BB cable routing, the cable pops out of the guide, which can cause weird shifting.
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N = '96 Colnago C40, '04 Wilier Alpe D'Huez, '10 Colnago EPS, '85 Merckx Pro, '89 Merckx Century, '86 Tommasini Professional, '04 Teschner Aero FX Pro, '05 Alan Carbon Cross, '86 De Rosa Professional, '95 Gios Compact Pro, '95 Carrera Zeus, ‘81 Masi Gran Criterium, ‘81 Merckx Pro, ‘89 Cinelli Supercorsa, ‘83 Bianchi Specialissima, ‘VO Randonneur, Ritchey Breakaway Steel, Rivendell Rambouillet, Heron Randonneur, ‘92 Ciöcc Columbus EL