Spurs never asked for it, it’s media outlets like yourselves that branded them with it in the first place.
As you say It was the top 4, Arsenal, Chelsea, Liverpool and United.
Then Tottenham and City broke into that top 4 regularly over a period of time so much so “The Big 6” was formed
This happened around 2010 so I had a look around to see which clubs have qualified for Europe most since then.
Man City - 16
Arsenal - 15
Liverpool 15
Tottenham - 15
Chelsea - 14
Man United - 14
Aston Villa - 4
Newcastle - 3
West Ham - 3
Leicester City - 3
Everton - 3