Assuming everything was within its limits since we can't prove otherwise and therefore a legitimate take-off, it's the captain's call whether to take-off or not.
Nevertheless, judging by the video, the precipitation seems far too severe to consider this take-off completely safe. Also, the crosswind might have been quite strong considering the open spoilers (also called airbrakes) caused by full left aileron being applied to compensate for the crosswind, which ultimately increases the take-off distance.
Furthermore, if it was a stormy cumulonimbus cloud, the take-off could have been delayed by half an hour, and the meteorologic conditions would have been much better in terms of precipitation (and it would avoid scaring all the passengers).
From my point of view the main problem with such take-off would be in case of a take-off emergency (like an engine flame out due to the intensive water ingestion) before V1, forcing the pilots to brake, risking a runway overrun due to hydroplaning.
They made it, but I don't think it was the smartest decision!