I filled out my application for Platinum on June 21 and received an answer today approving it. It said I will get the card within 15 days.

It is possible that they take longer for Platinum than they do for Gold, but I can't see why they would.

On the other hand, they did not ask for any proof of my Delta status. The delay might have been caused while they waited for an answer from Delta. Here, too, I have to wonder -- I currently have Delta Silver Medallion status, which only qualifies me for Sixt Gold. Until February, I had Delta Gold Medallion, which does qualify me for Sixt Platinum, so they may have asked Delta for further background on me.