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But he writes "and turned the oven up to 340 F". That's about 170 Celsius. The oven probably isn't very precise, but I doubt that's in the 250 Celsius range.

As an anecdata point, mine says it goes to 300. I get 230-250. Bread likes it hot.

I'm guessing he put the board in and then started the oven which took a while to preheat. Industrial reflow heats up the board gradually based on the solder paste manufacturer's suggested heat profile or experimentation/trial and error (e.g. when dealing with small pitch BGAs). 250C for even a few minutes would do quite some damage even to the fire retardant laminate and smell horrible.

He stated he was using a temperature of 340F, which is roughly 170C. Would that temperature also be harmful to SMD components when exposed for several minutes?

Doubtful. As a reference point, typical guidelines for baking out moisture (prior to assembly) are 24 hours at 150C.