Sarek's motivation for denying Michael's admission to the science academy sheds some light on his strained relationship with Spock.
I think Anson Mount's subtle portrayal of Pike adds perspective to the appearance in the Menagerie.
I found Ethan Peck's Spock to be a bit too melodramatic in places, although some of his bitchy exchanges with Michael work well to highlight his sense of humour, and foreshadow his relationship with McCoy.
The portrayal of a younger, more dangerous, Mudd works well. Roger C. Carmel gave Mudd a really sinister undercurrent which was masked by bombastic bravado but he gave you chills when the mask dropped for an instant. Rainn Wilson has played it with less melodrama and is quite a credible antagonist.
[SPOILERS] The notion that he may have found the robot planet from I, Mudd, years before they decided to imprison him is interesting.
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The real revelation for me though has been Mia Kirshner's portrayal of Amanda. I felt Winona Ryder was given nothing to work with. She was fluffy and written purely so that NuSpock could mourn her, giving the character (like so many of the women in the new movies) no respect.
Kirshner's portrayal on the other hand is great. She has a strength and sense of humour like Jane Wyatt's portrayal. We see a woman used to wielding influence as the wife of an ambassador; a woman who finds ways to cope on a planet where emotions are considered to be unsavoury, and a woman who can be quite innovative.
My main beef with Discovery has always been the technology. I don't mind a visual make-over for a modern audience, I don't mind alien tech being retro fitted for a few episodes, but don't try to retrospectively upgrade Federation technology in ways that affect the story. No casual site to site transporting, no wondrous new method of space (or time) travel, no fleets of battle ready fighter craft, no changing the size of the Enterprise for no reason etc. They tried to drag Trek in their preferred direction and it was like nails being dragged down a chalk board for me.
That said, in Star Trek Beyond, I cared way more about the fate of the Franklin than I did the Enterprise, so maybe it's just me being a dinosaur.