There's few things worst than someone that hasn't had the training and experience attempt to perform a thorough visual examination on a weld. I mentioned the mantra; Before, During, and After for visual examination already. the mantra needs to be taken to heart. The individual tasked with performing the visual inspection must have a basic understanding of the technology, processes, how to read drawings and understand the symbology used to specify the welds, a little basic understanding of metallurgy so they understand how the metal responds to the thermal cycles associated with welding and the significance and importance of the material specifications, the typical discontinuities found in both the base metal and the weld, and of course they need to be able to use a standard to determine the acceptable limits for the various discontinuities. After all, you don't want the individual to reject welds that are acceptable, but you don't want them to accept welds that don't meet the standard.
In many case, a good welder can be trained to perform the visual weld inspection. His experience and practical knowledge of welding is useful because he knows where the discontinuities are most likely occur and what most likely caused the problems. His knowledge and skill as a welder can be utilized to address the shortcomings so the overall quality can be improved.
Depending on where one lives, there are several recognized programs for training and certifying welding inspectors. You might consider sending a couple of people to the training courses even if they don't actually get certified. Many are available online using the Zoom platform. ASME, AWS, and several other organizations offer courses closely aligned with the training needed to perform visual weld inspection for very reasonable costs.
Best regards - Al