I think the stag served two purposes:
1. show the family's love of hunting
2. maybe it's too "arty farty" and I'm overthinking this, but I think it's a "parallel" or "metaphor" for Diana: Diana, a name of the Goddess of the hunt and how Diana is often described (even in her own words) as "lamb to the slaughter", "the huntress becoming hunted [by paps]" and all that.
So Diana's "initiation" into the family is helping shoot a major stag, during a hunt with Philipp, who crucially is another "outsider" (ie married in), but also the manager of the family in private and interestingly the last episode of this series and his... "threat" and advice...
I think the stag is a "representation" of Diana, ironically/fatefully killed by her (with her help, her direction and Phil at the gun).
If one wanted to read even more into it, it kinda even foreshadows the modern popular conspiracy theory of Phil allegedly ordering Diana's death.
So you can read multiple things into the stag bit, but the Diana/stag connection/mirroring is undeniable imo (the Phil conspiracy aside).
As soon as Diana showed up at Balmoral I knew she'd be the one to take the stag down.
I liked Olivia this season quite a bit, much better than last. HBC I always liked as Margaret.
The cast this series was altogether fanastic.
I don't get why they made such a faithful copy of Diana's wedding dress and then we saw so little of it? I was a bit disappointed by that.
Also disappointed by the portrayal of Camilla as his only mistress. She wasn't his only. He had more. Eg Kanga.
Also disappointed at Charles's portrayal as father, he was a better father than publicly known/shown. Much warmer and more caring, though yes, Harry's birth eg was scheduled to fit his polo...
I liked that they showed that the "Diana's mad" stories came from Charles's camp. That is, allegedly, very true. (In the same way women now are described as "crazy" for wanting texts back etc, you know the stories when guys admit their ex wasn't in fact all that crazy? They behaved like dicks and drove the woman into madness during the relationship? That.)
I think the protrayal of Liz as mother was spot on. It has been said for a long time that she wasn't the best mother, in the way shown pretty much. (absent, cold, never gets involved with anything, buries her head in the sand)
Some things were I think changed to compensate for the few episodes they had, but too much history. It's called "The Crown", so the focus is Liz and then around her a lot happened, obvs they can't faithfully recreate the whole Charles/Diana/Camilla relationship, that's why I think that even if in reality they weren't in touch for 5 years, they omitted this in the show, as he was allegedly during the honeymoon constantly on the phone to Camilla. [awfully phrased, but I think you get my drift]
I hated Diana's wedding anniversary gift in the 9th episode, I think they overshot it a bit with that one.
I wish in the last series they'd shown a bit more how Margaret and Tony's marriage fell apart, how vile they were towards each other.
Oh and one last thing: I hate how docile, nice and sweet they portray the queen mum. She wasn't like that. She gave Diana a massive bollocking for wearing a black dress during the engagament, as "black is only ever worn for mourning" and the cleavage was too tart-like. (that black velvet dress at the opear or some bday party or what it was)