The FA don't see any of the TV money because the Premier League is a separate body, you have the FA, the Premier League and the Football League (Championship, L1 and L2).
What they really should have done is find a way to fund a stadium build that wouldn't mean giving about 30% of its capacity away to people who don't care about the match or support either of the teams.
But hey, why use your brain when you can just not give a crap about fans...just brave-face it for 10 years, make up some rubbish about how the national team needs a new football centre. Hell, maybe they could even sit in the posh box with a a pair of giant inflatable middle fingers turned toward each end just to make sure everyone knows they don't give a crap.