I think that they actually want a tube running right to the stadium from what I heard from other stories about these alleged takeovers by middle eastern sheikhs .
The service line from SS to NP has been discussed before , one story I read by a reported expert on railways is that it's would need complete upgrading to be able to be used for passenger traffic and then there is the cost of building platforms at NP with all the disruption to the existing depot , all discussed before and deemed economically unviable by TFL.
From the southern end of the stadium its a 907 metre walk to NP where the Victoria line depot is , from the same spot outside the stadium to Seven Sisters it's approx 2.12 km, you would be saving a walk of around 1,200 metres it's not worth the money.
The lack of a tube station very close to stadium doesn't stop people attending the stadium in fact Haringey Council would positively be against it they want people walking down the High St spending money in shops and boosting the local economy.
Back to the podcast didn't listen to it all , expect it's full of the usual unsubstantiated drivel.
Lammy was wanting a station for years. So no haringay aren't against it.
Also yes it needs work, but it's not costing the same as tunneling for crossrail.