It's hard to say much about his interpretation of the evidence, since I don't know which papers he's referring to. However, there are some fundamental issues with his description of the immune response. Some of these are likely due to oversimplification for a wider audience; however, some things are simply wrong and could change the interpretation of these types of data.
In a typical, primary exposure to anything, IgM responses tend to peak around day 7, not day 30 as described in the video. IgG and IgA come up later than IgM in a primary response, but even they tend to peak at around day 14 in a typical exposure. During a second exposure, IgG and IgA do come up very quickly...usually more quickly than the IgM response of the primary exposure.
So it really depends on what the original data look like.