(For what it's worth - 29.53 * 300 = 8859 exactly, and 2000-01-06 + 8859 days = 2024-04-08, and I am quite sure there was a new moon on the latter date because of the solar eclipse.)
[If that constant is only a minute off, it ought to be good for a few centuries more, and I'm not planning on being around longer than decades, so probably good enough for my purposes.]
Honestly, I recommend people use the most accurate algorithm they practically can, even if it's overkill for the given application. This just avoids user confusion when different apps give different answers, even if it's not significant.
E.g., I have ported VSOP87 which gives the position of the moon and planets to sub-arcsecond accuracy to a couple of dozen languages. They are much, much longer than the "snippets", but are not impractical, and users won't notice much difference in application size or computation time. https://github.com/gmiller123456/vsop87-multilang