Best way to get rid of patio weeds (see picture)?

Chemical warfare, with a maximum cary-over non-selective herbicide. In plain terms, like Ortho's "Ground Clear", or "Triox", or possibly Round-Up Extended Control. The right stuff will last enough to prevent new weeds for the rest of the growing season. Skip regular Round-Up - it has minimum carry-over, so new weeds can pop next week.