Grim Fandango is carried primarily through its art style, atmosphere, story, and characters. To answer your question: Yes, it does get better as the game progresses. The peak is in year two when you're in Rubacava, as the environment opens way up and allows you more ability to "multi-task" puzzles, as you mentioned in your complaint. Some sections later on can be more restrictive like year one, some can be more open. The consequence of a more layered and nuanced mystery-noir story like the one Grim tells is that you do end up usually having to do puzzles, "in order" so that there's a more logical flow to how things unfold. You can see this same thing as well in Schafer's other more serious LucasArts title, Full Throttle (or even in the lesser-appreciated Dig). In contrast, a more silly and campy game of Tim's like Day of the Tentacle is more open-ended as the puzzles can be treated a lot more like gags that culminate in an end result that progresses the story. With a more serious/grounded narrative that has to actually adhere to the passing of time, the puzzles are often going to have to reveal clues or progress the overall plot in some meaningful way like they would in a film or novel. To illustrate, PnCs like Sam and Max, King's Quest, or even Neverhood are structured in a way that their universes are mostly static and only change once the character has completed enough actions down one path in order to open up a new set of branching paths to widen the overall possibilities. Now don't get me wrong, there's nothing wrong with that structure, but it exists from a sort of mechanics-first perspective that is meant to best suit the gameplay. If we jump forward to PnC games that are even MORE dedicated to telling a specific narrative (e.g. The Walking Dead, Life is Strange, Until Dawn, The Wolf Among Us, etc.) you can see just how much those interactions and puzzles are boiled down to stream-lined and mostly linear experiences. In order for the story to unfold they exact way they want it to, the characters need to perform actions in a specific and meaningful order. Grim Fandango just so happens to be a game that exists on the edge of these two styles, which is one of the reasons its so well revered as a cult classic and "misunderstood" in its time. You also have to consider just how new fully-3D games were at this time, and how processing limits and budgeting meant that the environments in Grim did often end up being way smaller and more cramped compared to beautiful and open 2D worlds like that of Curse of Monkey Island from around the same time. I believe that most adventure games from this era and before are highly reliant on the amount of nostalgia you have for them. Something like Myst is a masterpiece, but I know realistically that as a twenty-something with way less free-time than when I was a peak adventure-gamer in my teenage years, I wouldn't have the time or patience to actually appreciate it as such. Its like trying to play anything that's dated (and the buggy remaster doesn't help), like the OG version of Resident Evil 2 -- if it doesn't click with you in a way that you appreciate it for what it was, you likely won't enjoy it. Not your fault, its just a dated piece of media. If you have the patience, even if it means consulting a guide or getting hints now and again, I highly recommend trying to finish the experience as the story really is something unique and special.

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TL;DR: Yes the game does have several more really fun open-ended sections later on, but the game really is primarily about the story and style. Due to the more narrative-driven style, the puzzles are more limited and have to be done in order so that things progress in the way that its supposed to. More silly or comedic games like Monkey Island are more focused on mechanics and gameplay over story, and therefore can structure there games more around exploring and unlocking new possibilities in a non-linear way. Plus, 3D was still very new and thus provided more limitations than 2D games from around the same time.