The appraisal skill basically lets you guess (accurately or not) at the value of an item. As an in-game, hardwired skill, it basically gives the player a basis for bargaining with someone when selling treasure. This is mildly valuable, since if you're going to involve some bargaining then the seller has to have a basis for asking one price or another. How to handle this?
How is it a test of player skill as opposed to a test of the character sheet?
The job here is to involve player skill in some way (if it's highest quality) or make it an interesting, game-within-a-game test of the character sheet, or else there's no real point to it other than perhaps making a die roll to see how much the treasure (or whatever) actually goes for -- perfectly fine in a regular home-made adventure, a bit weak if you're charging for it.
Most likely this is addressed as an opportunity for a bit of roleplaying, to let the players bargain with an imaginary person, switching up the pace for a short time from die rolling. That's valuable if it's done right. How do you introduce this little bargaining session in a meaningful way?
Given the above, it seems like the optimal approach is either to just hand the players a number they think the item is worth (without needing to randomize it), or to give them a couple of pieces of information they can use in the back-and-forth of a quick haggling session. "The dwarf sees that the gem has a slight flaw." "The fighter notices that the horse has a slight limp." "The thief recognizes this as an antique from the Quoo-Am dynasty, adding additional value." Then they can either feel that they have to hide a problem (directing the buyer's attention away from the limping horse by pointing out the beautiful sunset as you describe the buyer starting to check the horse's legs" or little seeds for fun interactions like that. The focus isn't really on the value of the item, it's on creating a couple of seeds for fun (and SHORT) breaks in the action. Everyone remembers the "These are not the droids you're looking for" scene. Really that's just a traffic stop with a bit of interaction with the cops and a quick-thinking response. The goal with treating appraisals in this way is to create the "Not these droids" encounter, not to actually measure or set the value of treasure.
Handling it this way (examples)
As above. "The [character] notices that the [treasure] is/has [flaw or additional value]. A regular [treasure] is probably going to be sold for [their guess, just hand the number to them], but the [flaws or extras] might drive the price up/down to [top or bottom of negotiating range]. This creates a simple situation where the characters already know the approximate value of a regular item, and know how much their roleplaying could earn them.
If it's not worth roleplaying, just tell them the value so they can write it on the treasure list and move on to the action. It may be worthwhile to give them a value in the country and in the city, to give them a meaningful decision about where to go next.