You literally have a giant open world filled with so many more puzzles than any other Zelda, between the shrines and those in the overworld, that i still don't know how this can be seen as an issue. Also the main problem with the beasts is the size, not the puzzle design/mechanics. Ruta and Naboris in particular are brilliant, and so is Hyrule Castle.
Also not sure how writing is worse than previous Zelda, and nobody seems to mention how well directed the cutscenes are.
About the story, as i said in the thread about the dlc the problem isn't what is told, but how. The devs made the choice to tell you what happened 100 years ago from fragments of Link's memory that you only get to see in cutscenes, which makes sense from a gameplay perspective considering they went for a non intrusive narrative that didn't take you away from exploration too much (aka what people were asking vocally until last year), but this choice obviously has some drawbacks.
A large party of the story is in the past, some of the best characters are in the past, and you don't get to be invested in those things as much as if you had played them directly. Hopefully i'm right and the dlc2 will address this, in that case (and if it's done well) there wouldn't really be a problem anymore. The game could end up having one of the best stories in the series as a whole, because what it wants to tell is already good. It just needs to get you more invested in that.
Sorry man, I kept meaning to reply to your post and it was in the back of my mind, it's just been a busy several weeks....
Technically, if we put aside the Korok seeds (which are fun but have a limited number of puzzle types), or at least reduce them to their number of puzzle types, I don't know that the game features more puzzles than any other Zelda. I would think for example in TP or SS the combined # of puzzles across dungeons and the overworld is similar to those found in BOTW, but maybe I'm thinking about it wrong.
Many of the shrines we've encountered (about half of those in the game total) have no puzzles at all. Some of them have very clever puzzles that we liked a lot, specifically Speed of Light, and one dungeon-esque shrine at the end of the Goron Mine Coaster, Blue Flame, was memorably reminiscent of classic 3D Zelda dungeon areas. With that said, we personally didn't care so much for the multi-solution environmental puzzles available in many shrines. I understand this sort of puzzle is of great interest to some, it is just not our thing and we find more satisfaction in the traditional style of Zelda puzzles where there is usually one solution. So, for us it becomes an issue that despite the number of puzzles they don't offer the sort of puzzle we are seeking.
As to the design of the beast dungeons, I can only speak to our experience with one, as we've only played the Goron one so far. So, I'm glad to hear the design of the others is notably good. I'll look forward to that when we revisit the game, then. If the main problem remains the size, though, I'm glad to know that as it tempers my expectations a bit.
You make a good point regarding cutscene direction. I haven't given that enough credit but it is done well. I think the previous Zelda games have excelled at this also, but maybe I don't have the knowledge base to recognize an additional level of skill in directing shown herein.
As to the writing itself, in terms of the depth and characterization, I think it is not as deep as that in some of the previous games, specifically Majora's Mask comes to mind as a high water mark for the series when it comes to providing expansive depth to characters. There are so many NPCs that of the ones we have encountered, main story and sidequest characters, few have been memorable or had any complexity in characterization shaded into them. There are still some areas we've haven't gone to, but after 110+ hours we were disappointed in the lack of depth of characterization. The main story plotting itself mostly feels generic from what we have encountered (several memories and some of the main stories). I agree with what you wrote about the advantages though (and the drawbacks) of having much of the story told via flashback.
I also wrote some additional thoughts regarding our experience so far with the game
in this thread.
BTW, when I see screens and think of how much of Hyrule we've left unexplored, I do get some yearning to return....especially with the addition of the Hero's Path and the Korok Mask.