Harry's narrative in this series is very much aimed at US audiences whose media absolutely ignored the dead/injured from Iraq/Afghanistan (due to political reasons).
The UK media did cover it (maybe not as much as some would have liked), but The Sun and a couple of other tabloids did do a lot of stories. And a huge amount of campaigning alongside Help for Heroes and raised quite a bit of money.
So Harry is partly correct in what he says, but it will ring true mainly for US audiences. He also didn't correctly state it was Australian & US media only that leaked he was deployed overseas, UK media stuck to the agreement. The issue is US audiences will just assume he's exclusively meaning UK press. He's currently suing UK tabloids so they are going to pull him up on inaccuracies.
I would have liked a bit less focus on Harry speechifying as it interrupted the flow of the veterans stories at times (IMO), but it's a hard balance to strike, as without Harry there wouldn't be the same level of interest. Overall I think NF got it right.
As to the point of the OP and the story posted - it's true the focus of the story should be the veterans, any "salvaging" of Harry's reputation is entirely secondary. People have entrenched views now though, the doc will help as it reminds audiences of the "old" Harry that was the most popular Royal, but won't overall change things IMO.
Invictus is a hugely worthwhile cause, and anything that strengthens it is good.