I found the issue is not limited to anchor_sections, but also to other scripts that listens to the DOMContentLoaded event.
Below is an example of header-attrs.js.
By inspecting the rendered page, section div and the corresponding heading has the same class "rmarkdown", which should be removed from the heading.

---
title: "My First R Markdown Document" 
runtime: shiny
output: html_document
---

# R Markdown {.rmarkdown}

## Overview

R Markdown combines markdown (an easy to write plain text format) with embedded
R code chunks.

I also think KaTeX with runtime:shiny in my PR does not work due to this issue.

#1940 (comment)

Thanks @cpsievert for the report and thanks @atusy for the start of investigation.

We'll need to have a deep look into that - Rmd with shiny runtime is not always easy to debug.

By DevTools, we can see the real R Markdown content is inserted by the iframe element.
I think the insertion takes place afterDOMContentLoaded, and that is why event handlers do not fire if they listening to DOMContentLoaded.
In some case, we can workaround the issue by using async or defer.

Reprex

---
title: "My First R Markdown Document" 
output: html_document
runtime: shiny
---

```{=html}
<script>
document.addEventListener("DOMContentLoaded", function() {
console.log("DOMContentLoaded")
})
</script>

<script async>
console.log("async")
</script>

<script defer>
console.log("defer")
</script>
```

Just to note I opened a PR rstudio/htmltools#201 so that we can easily async/defer scripts.

So this means that any html dependencies used in R Markdown and using a listener on DOMContentLoaded won't be effective when using runtime shiny - Is that so ?

Yes... as far as I understand.

Ok, I merged your PR in htmltools (we were already planning on making that fix).

It might also be worth looking into just using $(function() {}) instead of the defer+DOMContentLoaded approach since that'd be more consistent with how do things elsewhere in Shiny & R Markdown

@cpsievert Thank you so much. I agree $(function() {}) is the robust way, however, it means anchor_sections and other scripts require jQuery. There are cases where people want to use html_document without jQuery (i.e., theme = NULL). I think it is nice if we can provide the functionalities without jQuery as much as possible.

@atusy was quicker than me 😄

Yes JQuery syntax is definitely an option. I am always trying to not depend on JQuery for new (small) scripts in order to maybe one day, have JQuery-free document. This is a wish from some member of the community for the R Markdown output produced. And indeed already possible with theme = NULL.

In shiny ecosystem, will you keep rely on JQuery ? I understand bootstrap 5 will not use it (by default).

In shiny ecosystem, will you keep rely on JQuery?

I don't see it going away anytime soon

Ok thanks.

For R Markdown, currently JQuery dependency is only inserted with theme != NULL. So it is a matter to decide with features powered by html dependencies with JS should be available for custom theming. (that advanced users are using from what I know)

I re-implemented dependencies with the defer attribute, however, there needs some more improvements to support runtime: shiny.
See #2046 for the details.