GP comment did not imply that half of elements are below average. It just said that _some_ have to be. By definition, we cannot have all elements be above average: at least one has to be below.
Ludicrously pedantic nitpick: this only applies to finite sets - consider the sequence 1,1/2,1/3,1/4,... (the harmonic series). The average (mean, median, mode[0]) is (depending on how pedantic you want to be) either 0 or 0+ε[1], but in any case strictly less than any positive real number, while every element of the sequence is a positive real number.
So it's not true by definition; it's a consequence of the basic sanity constraints that you're working with.
0: Strictly speaking mode only applies to continuous ditributions (ie, with a continuous probability density function) or fully discrete distibutions (eg heads vs tails), but 0 is the only (real number) x such that for any sufficiently small positive distance δ, the number of elements in x±δ is strictly greater than the number in x±2δ but not in x±δ (namely, all but a finite number of the inifitely many elements in x±2δ are also in x±δ).
1: Where ε is some surreal number[2] strictly less than any positive real number, but not necessarily 1/ω specifically.
Vanishingly few if any contain 11 or fewer yolks.
this is the pedantic part. Normal usage of average means "pretty much what I expected", which is what they are trying to convey, imo.
In any case, "average" does not mean "arithmetic mean." It means, mathematically, any of the median, mode, or mean or in standard vernacular, of typical or usual characteristics (i.e., close to mode).
You don't get to just arbitrarily pick a definition that fits your viewpoint and start going "all due respect" on other people.
In this case we're not in a mathematical context though.