Hi, Pete. There are a few observations related to this, not widely known although basic, and that includes your colleague. First, Conway gives a quick proof on page 142 of The Sensual Quadratic Form, including over the rationals.
Next, also Conway, the form (five variables) that he and Schneeberger found that represents all the numbers from 1 to 289, fails to represent 290, then represents 291 and on forever, he initially called Methusaleh. It is just a binary added to a ternary that represents the numbers from 1 to 28 consecutive, discriminant 29. However, for ternaries that is not the record. The form he called Little Methusaleh, discriminant 31, represents 1 to 30 consecutive. The theorem is in this material, as the conditions for a positive ternary to represent, say, 1,2,3,5, places strong restrictions on a partly reduced form. Kap wrote this sort of argument up several times, including a repeat in the unpublished 1996 Classification. It is quite easy. OK, Little Methusaleh and your result over the integers are proved on page 81 of The Sensual Quadratic Form
Finally, a positive form is anisotropic at the "prime" infinity. In Cassels Rational Quadratic Forms he shows global relations on the Hilbert Norm Residue symbol that show that any ternary is anisotropic at an even number of primes. So a positive ternary is anisotropic at an odd number of finite primes. Taken with the observation above that at least one number below 31 is missed, and a positive ternary fails to integrally represent an infinite number of positive integers.
I will look up some of my tables and fill things in. Note that some of this is discussed in an early article by William Duke, 1997 Notices, but he mistyped the form with discriminant 29.
Let's see, Conway and Schneeberger probably had an acceptable proof of the 15 Theorem scattered about, but it never got put together. Bhargava was looking for diversions from his own dissertation, Conway mentioned this in passing. Bhargava showed the fundamental result that one of these forms must have a regular ternary as a sub-form, thus the project became a careful inspection of my paper with Kap on all possible regular ternaries. Also, correspondence between Kap and Bhargava first revealed some important errors in Magma relating to calculating the spinor genus, and hilarity ensued.
EDIT: thinking about the history question, it is quite possible that this result was never written down as a separate proposition, by Gauss, Legendre, etc. The reason I suggest this is the great weight placed on positive ternary forms missing certain "progressions," in the language of Jones, Dickson, other early books. So, in Jones, chapter 8, we read "Thus there will be a finite number of arithmetical progressions of this type" of numbers not represented by any form in the genus under consideration. Not much motivation for proving that a form misses at least one number if you are going to quickly show that it misses an entire arithmetic progression.