Overall, I loved the finale and I really love this show.
I've seen a number of criticisms levied against the finale both here and on various other forums and subreddits. Frankly, none of the things raised as issues in these criticisms have bothered me in the slightest.
I don't care that Luke was another Deus Ex Machina or that the show has relied on them a few times. As long as they make sense within the story they're trying to tell I am fine with it. Furthermore, I think in a sense they're used in a way to propel forward the idea that as long as Grogu is with Din Djarin, his life will be in danger. Grogu will always be in both physical and spiritual danger while he's with Din Djarin and only a Jedi is capable of alleviating those problems. At some point either Din (or some other entity) will not be able to save the child from physical harm. At some point, without training, the darkness within Grogu could get the best of him.
If there is anything that Luke's entrance and conversation suggests is that he is exactly what Grogu needs to have a chance. I think the show did a remarkable job of keeping that focus. It's precisely why the oft criticized lack of introduction or reaction to Luke by Din and company is not something I even thought about as being problematic. Dave Filoni and Jon Favreau know exactly what they're doing in this scene. They're setting up Luke to be something to behold. The hero that many fans were hoping to see in The Last Jedi and didn't (well, not in the way they expected anyway). I am not among those fans, but I loved his appearance regardless. When all else seems lost and the only thing on Din's mind is protecting Grogu, the response to Luke is precisely what I would expect.
The first reaction we get is from Cara Dune who sarcastically questions the value of this single X-Wing. Next, Grogu begins to connect with and sense the presence of another being like him. Fennec Shand is confused as to why the droids would stop. Moff Gideon looks like fearful of the unknown that has just arrived. Bo-Katan exclaims with a little fear and a little hope "A Jedi?!" Moff Gideon now realizing this could be his end tries to go out on his own terms and is knocked out. After blocking Gideon's fire, Din watches Grogu as he turns his attention / reaches out to this Jedi. They all watch as this Jedi eliminates this enemy that was all but certain to destroy them. Grogu clearly trusts this Jedi (and Din trusts Grogu) and points to the doors to be opened. A healthy amount of fear of the Jedi is evident (especially because one hasn't been seen in so long). At that moment there is only one thing that matters and that's Grogu's safety. It's clear he can connect to and trusts the Jedi. It's clear this Jedi can protect Grogu. After all that has transpired, Din Djarin and company don't need to know anything more or anything less. There doesn't need to be any pleasantries. The child is safe. That's it.
I have no doubt that Din Djarin will likely learn the name of the Jedi (perhaps from Cara Dune) and will reunite at some point with Grogu. But at that moment, I simply don't think it mattered. Not to me anyway.
Regarding, the Darksaber situation. I think it all tracks pretty well with established lore. So much so that Sabine's mother actually asked her where she got the Darksaber. When Sabine replied that she found it in Maul's lair and not by besting him in combat, her mother said "then you have no claim to it." It's clear that there is a long-standing tradition that one can only make claim on the Darksaber if one defeats the previous wielder. Sabine had no intent of using it to rule Mandalore herself and by the time she passed it onto Bo-Katan, Maul (it's true owner at that point) was dead. Bo-Katan probably had the best claim to Mandalore at that time.
My guess is part of why Bo-Katan is taking a hard stance on this is because SHE WAS bested in combat by Moff Gideon during the Imperial response to the "Heroes of Mandalore" episodes in Season 04 of Rebels (a story we haven't seen yet) and she feels like the only way she can legitimately reclaim it is by adhering to said tradition.
I really enjoyed the episode and it did nothing but make me more excited for the future. Going forward, I don't really want The Mandalorian to be about bounty hunting or anything like that. I want it to be about what it means to be a Mandalorian and reclaiming the planet. But I also hope there is a check-in on Grogu from time-to-time. I am really looking forward to all of the other shows that will connect into this particular part of the timeline.