It does look like it's the beige connector, judging by images such as this:
https://i.imgur.com/5bmLvyn.jpg
If that connector is clean, perhaps try giving the blower a tap/spin as Marty suggests. Brushes certainly sounds a plausible explanation. If it starts working after giving it a nudge, I'd suspect warn brushes is your problem.
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Mark
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2002 FE S8 Ebony Black Pearl
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Cars Owned:
The Tesla Era: 2020 Model S Performance Ludicrous+ (present)
(Black, with all black premium interior and carbon fibre décor, 21" sonic carbon twin turbine wheels and FSD capability)
The Audi Era: '97 A8 4.2 (Ming Blue) --> '96 A8 4.2 QS (Dark Green) --> '02 FE S8 (present)
The Citroen Era: '84 BX 1.6 RS --> '89 BX 1.9 DTR Turbo --> '94 XM 2.0L Turbo --> '96 XM 2.0L Turbo Exclusive --> '00 Xantia Activa 2.0L Turbo
The Banger Era: '76 1.2L Lada VAZ-2101 (Ruski Fiat 124) --> '80 1.7L Morris Ital HL, finished in Ermine White and Rust