This isn't a command to keep vows or not make false vows...but to not make vows at all. Otherwise He wouldn't have been using language like "But I say to you", "make no oath at all", and "But let your statement be, 'Yes, yes' or 'No, no'; anything beyond these is of evil."
If it was about 'false vows', Christ would have left it at "You SHALL NOT MAKE FALSE VOWS, BUT SHALL FULFILL YOUR VOWS TO THE LORD." Instead it starts there and says "But I say to you".
If it was about 'false vows', Christ would have left it at "You SHALL NOT MAKE FALSE VOWS, BUT SHALL FULFILL YOUR VOWS TO THE LORD." Instead it starts there and says "But I say to you".