"Satire" isn't quite the right word for what I think Shin Godzilla is trying to be. "Humanizing" might be a better word. Not that there aren't satirical elements in it, because there are. But, it's not really a take-down of government bureaucracy per se. It's really more about showing the behind-the-scenes decision making that goes into dealing with disasters. It's about showing how politicians really are. And all the technical mechanics that go into disaster relief.
You have to remember, this is a Japanese movie. And Japanese culture is a bit different than Western culture. In general, Japan is a bit more polite and respectful, at least in public. And Japan isn't quite as cynical (or outright hostile) about their politicians, as we are in the USA.
I get the impression that in Japan, their leaders are kind of put on a pedestal. They are in charge, and you are expected to show them some level of respect. Again, at least in public.
There's also the Japanese concept of the "public face and private face". Your "professional" outward appearance you show at work, and your true "inner" thoughts that you might only show to those very close to you.
So, what this movie is doing is possibly showing us the "private face" of politicians. The mechanics of how they deal with disasters and surprises. How they really are, behind all the media and propaganda. It shows that they are flawed. And that they are human. But, in this movie, they all end up at least trying to work together for the greater good. Shin Godzilla is actually not that cynical of a movie. At the end of the day, all the politicians do end up getting out of each other's way in order to achieve their goal. This movie shows that the system CAN work. It's less about showing that politicians are incompetent, and more about stripping all the public veneer away to show us how human they can be. It's about showing that our leaders are a lot like us.
If this movie was made in the USA, I doubt that would be the case. If this was made in the USA, all the politicians would have been revealed to be cackling villains profiting off all the destruction, teaming up with a bunch of polluting mega-corporations for kickbacks, and Godzilla would have atomized them all with fire breath. And then there would be a group of schoolchildren cheering their death.
I think this movie doesn't connect with a lot of audiences outside of Japan, because it IS so culturally Japanese. We're just not used to politicians being depicted as...non-evil. Not exactly selfless paragons of virtue, but generally they're genuinely trying their best, in this movie. I think Shin Godzilla is trying to be more "The West Wing" than "The Great Dictator", if that makes more sense.
But, that's the thing about Shin Godzilla. It's one of the "deeper" Godzilla movies. You CAN dig into it, and find some things to discuss about it. It's a slightly deeper affair than most of the Godzilla filmography. You actually can have differing interpretations of it, and actually have a proper discussion about it. So, I feel like it is worth watching. Even if it is kind of slow. And slightly odd.
But, it's got it's fun moments. Like how learn that humanity's greatest strength is the ability to work together. So...Godzilla adapts to that and ends up as a pile of bodies fused together. It makes me imagine that a sequel would be more like "Secret Invasion" or "Invasion of the Body Snatchers". Which could be a fun movie, even if it wouldn't really feel like a Godzilla movie anymore, if they went that way.
I'll admit, it's not my favorite Godzilla movie. I DO tend to favor the "fun" side of Godzilla. The cheesier, popcorn action movie side of the franchise. But, I think I respect Shin Godzilla. It's not one I really have a burning desire to rewatch. Outside of maybe a few scenes. But, still, if you're willing to look into it a bit more, you might find something worth discussing.
But, the good news is, now you can get into the "fun" side. And I really think that's where Godzilla shines, personally. If you really want to sit down and watch a good chunk of them, my recommendation is to go with the Heisei Era movies from the 80s and 90s. But, that's up to you.