The cover certainly can be used for commercial purposes: See the debate around Heino and his Cover-Album of Tote Hosen Songs. In your example the band was allowed to perform it in public (commercial usage here) and they could probably have pressed a recording and sold it. You need a license and need to pay royalties but that license cannot be denied to you (basically registering the cover with the GEMA is sufficient). Note that this applies to strict cover-versions only and only for songs registered with the GEMA - no major edits permitted.

If that still holds true if the cover is used in a documentary is a separate question. Also note that sampling is not covered by that paragraph (see Metall auf Metall).