The wielder of weapon with magic weapon special ability grayflame should be allowed by the magic weapon special ability to take a swift action to charge the weapon by expending a use of the supernatural ability channel energy. While the weapon special ability could've been phrased to make how to use it more obvious, limiting the magic weapon special ability's use to creatures that can already take a swift action to channel energy into a weapon makes the magic weapon special ability nearly useless. (Even the question's example of the feat Channel Smite doesn't specifically allow a creature to channel energy into the creature's weapon!)
Channeling energy into a weapon is a thing for a couple of different weapons—there's also the void scythe, for example, although how to use that weapon is more obvious from its description.
Note that when the magic weapon special ability grayflame appeared in the Advanced Player's Guide (2010), Pathfinder was barely a year old and still drawing a lot of its inspiration from the game on which Pathfinder is based, Dungeons & Dragons 3.5e. In that game's Player's Handbook, players are told in Other Uses for Positive or Negative Energy that "[p]ositive or negative energy may have uses other than affecting undead [which is typically all channeling energy did in that game]. For example, a holy site might be guarded by a magic door that opens for any good cleric who can" channel enough positive energy into it (160). Likewise, an example in that game's Player's Handbook of the skill Use Magic Device has iconic rogue Lidda finding "a magic chalice that turns regular water into holy water when a cleric or an experienced paladin channels positive energy into it…" (86).
In a similar fashion, a player coming to Pathfinder from D&D 3.5e in 2010 would've likely automatically assumed the magic weapon special ability grayflame allowed the weapon's wielder to channel positive energy into the weapon, that being just another use of channel energy like as in that other game. However, nearly a decade on, I totally agree the description of the magic weapon special ability grayflame could stand a revision.