Why is my custard eggy?
With the full recipe, it does not seem to me that the egg ratio is particularly high.  Most recipes I have seen and use tend to be around 2 eggs per cup (US), and yours is 6 for about 2 1/2, so slightly higher than what I have seen as about average, but well within range.

To me, egg custards always have some amount of egg taste, which is pleasant or offensive according to taste.  When skimming the video given (had no audio available, so I am just going by visuals) I felt they were adding cream mixture too quickly to temper the eggs without cooking, but that is opinion only.  If the hot is add to quickly though, it will scramble the eggs and give a much stronger egg taste.  One technique is to sieve the custard after tempering to remove any solidified egg, and some are happy with this result.  To me, it removes the solids, but not the flavor.

I will give a personal critique of the video, I frankly have never heard of breaking a custard and expecting it to come back together in a reasonable tasting form, but the video maker obviously disagreed.  I can only say that I assume they are happy with a stronger egg taste than many of would be.

If you wanted to experiment, I would think dropping 1, maybe 2 eggs might still give you adequate setting power, but going all the way to 3 would seem too big a cut.  Try experimenting, reducing 1 yolk at a time and see if you like the results.  Also try slowing you introduction of the hot cream mix to the egg and see if this helps.