No, the Kindle cannot handle ePUB and the toc.ncx is very specific ePUB. Some converters from ePUB to mobi will convert the toc.ncx to an inline TOC, but not all (as you noticed).
In this case you need to create a toc manually as XHTML file. In Sigil that means a new text section with links in them to the headers. You need to identify this xhtml file as being the TOC. You can do that by right-clicking on the file and select the TOC entry.
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