Should singlehtml provide search utility?

[miketheman]

A question that seems to have almost been asked before, but never quite directly.

From its inception, the singlehtml builder declares non-support for search.

The search box was omitted from sidebar in #1147 and a request for indices in singlehtml is seen in #592.

This leads to such behaviors in templates to prevent the search box appearing in singlehtml mode:

{%- if pagename != "search" and builder != "singlehtml" %}

Is there anything inherently incorrect about being able to use the same search mechanism on regular html and singlehtml, or was it simply never built when introduced 6+ years ago?