An intelligent "layman" might watch that video and declare, "Math proves that 1 + 2 + 3 + 4 + … = -1/12; yet this is obviously wrong. Those mathematicians don't know what they're doing." Now, I do not know such a layman, but you must acknowledge this might happen. And this is the cost of the little lie permitted in the video you've linked.

If I wrote a "how to program" tutorial in which the code did not compile, you would likely berate my inept tutorial. Please allow mathematicians the same grace.

Nevertheless, that seems to be how people understand the video. See, e.g., the person whose comment I was replying to.