Watching the beasts was cool indeed. It reminded me of Majora's Mask, where you can always see the impending doom above your head, and the townsfolk also references it. But whereas getting to the moon was actually a very weird and unique moment, Divine Beasts felt too same-ish for me at the moment of tackling them. They also didn't allow much room for freedom beneath their dungeon structure, on detriment of Breath of the Wild's philosophy, but also on the series' general way of designing dungeons. The same goes for shrines in this case.

Taking a different path doesn't make Zelda games bad. Pretty much every Zelda iteration has taken a different path, or added a design twist to its foundation. It's just that the strongest elements of Breath of the Wild as a game itself struck me as neglecting the elements of what made other Zelda games great, just like the Resident Evil 4 example I provided to Veknoid. Resident Evil 4 took the franchise in an entirely different path, and for many people it was the best RE game, but I always felt the franchise was never truly the same after it. This doesn't necessarily has to be the case with Breath of the Wild, as it maintains some of the franchise's staples still. I'd love a hybridation of both models and see the result. Maybe it doesn't mesh as good as I would be expecting, but one never knows.

Actually you're a bit off here as Shrines allowed for a significant amount of freedom in how many of them are tackled I suggest you watch several speedrun categories as they show case this even the beasts themselves offered more freedom than what you'd find in other Zelda game's dungeons.

What made other Zelda games great was the adventure the player had not what you're suggesting, what you're talking about is structure which by the time BOTW had arrived had become the thing limiting the series appeal and progression this also goes for RE4 as well. 

By the way, here's some thoughts I wrote a while back about how I'd personally improve this new Zelda formula for the next game, seems fitting for this thread:

- Bring back traditional dungeons (At least 8 of them. 4 dungeons isn't enough for a map that big).

- Less shrines (40-50 at most), but they're bigger and don't share the same visuals anymore. Make the puzzles actually challenging.

- Make exploration more rewarding, because tbh once you realize that all you'll find are either Koroks or Shrines (Or disposable weapons), it becomes rather boring imo.

- Make voice acting optional if possible. I think Zelda worked better without it, I liked reading the story at my own pace and it was part of the charm for me. Light BotW spoilers: The flower lady cutscene wouldn't have been half as funny to me if it was voice acted, the exaggerated reactions are what made it good.

- More enemy variety. Each area of the map should have it's own exclusive enemies (Like Skulltulas and Poes in dense forest areas, for example).

- Less "copy-paste". What I mean by this is that I want more unique content in the overworld. Finding a labyrinth doesn't feel special when you discover that there are 3 more of them at the other corners of the map, only slightly different. Same thing for enemy camps.

- Main menu: Bring back the title screen, music (Fairy's Theme) and more save slots.

- Put music everywhere again. To avoid music getting annoying/repetitive, just make longer and/or dynamic themes (Kinda like how Mario music changes on the fly when going underwater). For example, if you have a theme for Hyrule Field, make it change depending on the weather/enemies/time/riding a horse/etc.

- Make the map a bit smaller, but add caves and underwater sections to the game.

- Completely revamp the weapon durability or get rid of it (Shields aren't damaged when shield surfing, weapons last longer, make it possible for us to repair them with certain items, etc).

- Give us the green tunic at the start of the game again. If people don't want to use it they can simply wear something else, since armor is likely staying for the next game.

- Good non-shrine quests, since most of them were just "Bring me X stuff and I'll give you something in return".

Of course this is all my opinion, sorry for the giant wall of text =P

The reward from exploration is not the trinket at the end, it's the journey. To make exploration more rewarding the landscape needs more variety, more surprises along the way, hidden valleys, caves, etc like you said. Better enemy variety helps with that as well.

Less copy paste and more unique content is the way to go. That's a lot to ask though, no other rpg has kept me busy for 170 hours so they did a lot right. But indeed, a lot of shrines felt like filler and you don't need 3 rows of hearts so they became quite pointless as well in the end. I have the same feeling atm with the moons in SMO, a lot of filler and I quit at 570 ish, are there really 999 in the game???

I've personally tried to sequence break shrines myself, but only found myself being able to do so in few shrines (especially those that use electricity, and you can cheese them by using metal of your weapons). Often I would just push through creative outcomes from the forced mechanic rather than freedom myself, such as using the gamepad ones and turning mazes upside-down to have a estable platform rather than finding the correct path. I've seen some speedruns, where they use the shield gliding + bomb explosion area to reach unreachable places, though I never use speedruns myself as a metric of anything because those players have perfected every single mechanic/exploit from games to achieve so. That being said, I've seen the any%, which surprisingly don't really use exploits but merely reach Ganon under 30 minutes by using the game's mechanics in their favor, and it's something I commend the game for.

And the adventure given to the player in previous Zelda games certainly include tackling dungeons; a game's structure is also how the game presents the adventure to the player, and vice versa (that's how I see it). Breath of the Wild's structure is a huge departure from previous Zelda games, which makes the adventure palatably different because of it. I just found it contrasting to the usual adventure/structure I was used to because of the points I made in the post. I put the Resident Evil 4 example because I always found that game was a bad Resident Evil game, despite being an awesome game on itself; I understand the Resident Evil formula had become stale by that point, but I just didn't like that it sacrificed most staples of the franchise in order to renew and broaden the franchise.

EDIT: I gotta say, some of the Main Quest speedruns are pretty amazing to watch as well.

Being used to something doesn't mean its a good staple the structure in the series was a limitation as if you removed the dungeons from all the previous games only SS and maybe WW would have some semblance of what makes it good the rest would be pointless this is why the structure had to be overhauled as the dungeons were the games Hyrule was just a stop between them. In BOTW Hyrule now is the game the player now no longer needs to get to a dungeon to enjoy the concept of the adventure or require a said item to open up and progress the game they are given a definitive adventure in which the structure lets the player themselves dictate how the game plays out in the world.

That is what imo makes BOTW the best Zelda game as now it feels much more like a proper true adventure within Hyrule where you organically come across villages, shrines and such in a quest to find a solution to the problem as opposed to a sort of dungeon crawler where you're guided from A-B due to a limited structure and the world outside doesn't really matter.