The search query input field on Kagi.com disables spellcheck through the inclusion of the textarea tag attribute spellcheck with value false:
<textarea ... autocapitalize="off" autocorrect="off" spellcheck="false" autocomplete="off" ... name="q" ...>
This prevents usage of device spellcheck in effort to self-correct our own typographical mistakes in queries entered.
I'm not aware of the reasoning behind this choice. I am interested in learning that reason or otherwise, being able to utilize device spellcheck.