How to support interactive content like `<input type=search>` in a menu?

The menu elements explainer (which I'm currently rewriting) mentions supporting <input type=search> a few times, as an example of other interactive content that can be a sibling to <menuitem> elements inside of <menulist>, and maybe even <menubar>.

This prompts at least three questions:

• Should we even support this? I don't think it's a common pattern on the web, but some heavier applications like Google Docs support it. Should we leave it to them to support? If we decide to support it natively, how far should we go? Should we consider what we discussed with combobox, which is maybe to support some native search mechanism out-of-the-box that searches through menu items in all of the popover-linked menu lists?
• What are the accessibility implications of this? Are screen readers generally compatible with encountering an input element in an application menu? macOS does this natively, so presumably the accessibility story is sane and well-thought-out.
• What about other interactive content (besides links of course!)? Do we need to do anything special if other interactive content is embedded in a menubar or menulist?

Start with the menu elements explainer mentioned in the issue, especially its references to input type="search" inside menulist and menubar. Review the accessibility implications and related interactive-content behavior before proposing a direction; done would require a decided scope and corresponding guidance, but the issue currently leaves that unresolved.