A cheaper rail+tube route is Stansted Express to Tottenham Hale, then
Victoria Line to Green Park, then Piccadilly Line to Heathrow - about 2
hours in all, and the interchange at Green Park is inconvenient if you
have much luggage.

That interchange is inconvenient even without luggage. A better interchange
would be at Finsbury Park where you can just cross to the adjacent platform.
The Piccadilly Line does have more stops in central London so can be more
tedious but actually only takes a few minutes more.

G.