First off I'll note that the linked article doesn't actually claim that
Triana DSCOVR is the first spacecraft since Apollo to see the whole daylit side of Earth. The relevant quote (emphasis mine):
DSCOVR’s two Earth-observing sensors will provide views of Earth that have only rarely been available before. [EPIC] will view the entire sunlit side of Earth at once, providing images reminiscent of the iconic Blue Marble photograph taken by Apollo 17 astronauts in 1972.
David has already pointed out that L1 is far beyond the Moon's orbit, and that the geometry of the "blue marble" photograph is much closer to that of images taken from geostationary orbit, like this recent image from GOES-EAST:
The GOES program traces back to the SMS series of satellites, meaning that we have had nearly continuous full-disk coverage of the Earth since 1974, only two years after the "blue marble" photograph was taken. There are two slight caveats: first, the GOES/SMS instruments have always focused more on the infrared spectrum (the current block has only one visible-wavelength band, and the next-generation GOES will have only two) so the images are not in true color. Second, they only produce fully-illuminated images around the equinoxes, when the intersection of their orbital plane (the equatorial plane) and the ecliptic plane lines up with the Sun. (This is in fact the definition of the equinoxes.)
However, there is an additional major hiccup: it turns out that geostationary satellites don't capture "the entire Earth in a single frame!" The current generation (which are 3-axis stabilized) use a two-axis steerable mirror to scan across the entire disk, while the early spin-stabilized GOES satellites used a one-axis steerable mirror to scan North-South and the rotation of the satellite itself to scan East-West.
It turns out that it is not unusual for departing spacecraft to take pictures of the Earth, simply because humans like looking at their planet! (Or because it makes a convenient calibration target.) However, most spacecraft leave Earth roughly parallel to Earth's orbit, so they end up seeing a partially-illuminated disk. MESSENGER and Galileo came close, but were still far off from the correct viewing angle. You are correct that we usually don't send spacecraft on orbits towards the Sun, and those that we have, have all been solar-observing satellites that were simply not equipped to take pictures of the comparatively dim Earth.
Here is LRO taking a picture of Earth during a solar eclipse in 2012! This picture is black-and-white, but it is as close to fully-illuminated as you can get (except for the shadow of the Moon!) and it was taken in a single frame. (LRO has captured other, shadow-free views of the Earth, but I wanted to share this one in particular.)
In conclusion, I think that the article's claim that the DSCOVR's view of earth is unique is mostly "marketing," unless we disregard mosaic images as you suggest.