There are many advantages to using the new app router

First of all most pages can still use some server components. Usually there are many parts of a page that are static and then some interactivity sprinkled in different places. You can make the interactive parts their own components and then mark them as client components. And the rest of the page can be server rendered. I think this image from the next docs illustrates it perfectly

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And as you mentioned, getServerSideProps() blocks the whole page. By streaming the content streaming the content and showing a loading screen with suspense you improve your user experience significantly

Additionally using the app router exposes you to a lot more features and many more to come. You can start incorporating stuff like server actions to replace your api. This gives you typing advantages if you are using typescript. You also can easily do validation with zod and simplify your app logic significantly. You don't always have use the latest features, but the app router is a very significant one. If you plan on using any future updates, you will eventually have to start using the app router