If you look at the top 5 most dominant teams on this list they are Man Utd, Man City, Chelsea, Arsenal, Liverpool.
They all have the top percentages for possession and tackling. Heading seems less important, so long as you are not bad at that.
I don't think the evidence is we are especially bad at heading. We don't concede a lot of goals to headers in the PL, although our air dominance here may be a statistical anomaly due to the Benteke Effect.
What we lack, as you suggest, is a physical presence in midfield. Our midfielders can pass but they can't tackle.
We shouldn't over-exaggerate the physical though - look at this.
What I find incredible, Swansea, which is a midget team, wins 79% of tackles, whilst Tottenham, which is a team of giants wins only 74%.
How does AVB restrict opponents to so few shots at goal whilst being so bad at tackling and weak in the air (despite their height advantage over most teams - tallest midfield in the PL!)?
Who is Swansea's defensive midfielder who wins all these tackles? They ain't got one.
Swansea's players may be playing a more effective defensive system to us, where they pick and choose tackles better.
They don't dive in but only tackle if they are definitely going to win the ball, otherwise they work with each other to shepherd opponent players to the wings where they are less dangerous.
This would mean the Swansea midgets will have attempted fewer tackles than other teams. (From a quick glance at the stats again - yes this is true!).
So it's not just a question of physicality. A big physical player who can tackle would help but as Swansea shows, it's not necessary. It's organisation.