If anyone needs a ‘proof’ of holding a current account with Monzo you can ask the helpful guys on the CA app chat to generate you one and they will email it to you as a PDF.
I needed it for a grant payment but I can imagine mortgage applications may require similar ‘proof’ of your current account. The letter includes a note that full bank statements are not available yet as they are “finalising designs”.
Once the current account is rolled out to everyone the Monzo team may want to consider sending everyone an initial welcome letter by post that confirms your account details to save having to request one. I guess this might be the same letter that the debit card is attached to?