[docs-infra] Fix page change speed regression

The problem

The page transition speed between pages is about x3-x4 too slow.

Open https://mui.com/material-ui/react-button/ and navigate to the Button Group page. It takes about 200ms to perform the transition:

According to https://web.dev/articles/inp, this is very borderline for my high-end MacBook Pro. Google Search will use a new performance ranking metric, which we don't do well:

https://search.google.com/search-console/core-web-vitals/summary?resource_id=sc-domain%3Amui.com&device=2#inp-table

The goal

Compare the experience to https://www.radix-ui.com/primitives/docs/components/alert-dialog to move to the Accordion page:

It's x2-3 better.

Same on https://ui.shadcn.com/docs/components/button, it's at about 65ms.

So as a developer, my first impression is: Material UI docs feels slow, then surely MUI System is not performant, I'm not using it, I want me end-users to have a great UX. To be fair, we can't really compare Material UI docs page to others, we have a lot more content on the page.

It is to be noted that it seems a regression, https://v4.mui.com/components/button-group/ is faster:

Part of the solutions

• Layout. We don't use layouts like the comparison I raised about. https://github.com/mui/material-ui/pull/38394 showcases the benefits of using a layout using the Next.js Pages Router. It helps a good amount. Same in https://github.com/mui/material-ui/pull/38050#discussion_r1268280912, https://github.com/mui/material-ui/pull/36732.
• Emotion. It's slow, we spend a lot of time doing style computations
• ?

Context

This issue is meant to be closed with insights gathered after the research task

Search keywords: docs-infra, page load

Start by profiling navigation from the Material UI Button page to Button Group and compare it with the Radix and shadcn examples. Review the layout discussions in PRs #38394, #38050, and #36732, then follow the linked research task. Done means identifying the regression's causes and documenting actionable insights or fixes.