Vim's :open does not tab complete, what does?

That might be because :open is not the command you're looking for. Try :edit or :e for short. Also try :help :open and :help :e to see, what the commands do. :e has tab completion.

Use :enew if you want to create a new empty buffer.

Use :edit filename if you want to edit a specific file in place of the current one.

Use :new if you want to create a new empty buffer in a new horizontal split window.

Use :split filename if you want to edit a specific file in a new horizontal split window.

Use :vnew if you want to create a new empty buffer in a new vertical split window.

Use :vsplit filename if you want to edit a specific file in a new vertical split window.

Use :tabnew if you want to create a new empty tab.

Use :tabedit filename if you want to edit a specific file in a new tab.

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Use :help :command if you are unsure about its usage.