Not really sure why is spying supposed to be a domain of hunters? Hunters don't spy per se, and the spying at Conclave couldn't be anywhere close to cartoonish walking on tiptoes and hiding behind pillars as if they're hunting an animal prey.
Plus, hey, look at real life - in spy movies we need all those top agents and advanced tech, while in reality the Panama Papers were carried out causally on a cheap memory disc
All they need is not bring too much attention to themselves, which is generally what most elves are pretty good at.
Anyway, I'd say that mages, and especially Firsts would be suited for spy missions as well as others, if not more - they usually take care of things like dealing or contacting strangers and know something about outside customs or politics (might use it to soothe out nerves or be less conspicuous), probably know how to read and write better than most (correspondence and interesting documents) and even have a legit reason to be at the Conclave, unlike random non-mage elves with Dalish tattoos - after all, they're mages, and the Conclave is in large part about mages.
Because the clan relies on hunters for all tasks that involve going outside.
There are rare exceptions, of course, but generally speaking the Keeper's apprentice wouldn't be the first choice for that sort of work. And it is funny that you speak as though observing from a distance were something spies never, ever, ever do.
Not to mention that the Dalish aren't Orlesian. If you imagine them taking a bard's approach, that wouldn't be accurate.
True, it is fitting for a mage to investigate mage affairs. But clan Lavellan wasn't trying to solve the problem. They just wanted more infirmation on what is happening and how it may affect them.