What should I be doing as a scout in the opening seconds of an arena match in Team Fortress 2?

You're not wasting the speed of the scout in arena. Running into a group of 12 people is pretty much guaranteed death.

Scout in arena is exactly as you're doing. You need to wait for people to spread out and rely on your teammates audio cues and voice communications to know where to go. Listen for sniper calls or watch the death tracker for sniper kills and then go pick them off. Pick off stragglers who might be retreating to find health after a larger fight. Arena means that every death is a tip in balance, and as you're the MOST fragile player, you've got a lot of liability on your shoulders to ensure that you don't waste your life right off the bat (no pun intended). Use your double jump to your advantage and consider using the FaN to get to good ambush / hiding spots.

Scout in arena can also be effective bait. Get that demo to chase you around the corner into the hands of a heavy/medic combo. A lot of people will tunnel vision and auto target scouts due to their low health.

To put it simply, if you're playing scout in arena, you're playing more like a spy in any other match.

Edit: Adding in a bit about defensive play.

You can also treat scout in arena like a 6v6 competitive match as the defensive scout. Your job is to protect the medic who will likely be glued to a soldier or heavy early on. If the offensive target (heavy or soldier) is an attentive player, the FaN is a great choice for this due to it's stopping power, buying them a few seconds to turn and destroy (assuming they're not dead already). If they're not terribly attentive, your normal scattergun will do the trick and you're just going to be trying to kill and dodge.

Lastly, while I don't use it often (I love me some pistol action), you can use the Bonk! Energy Drink and park yourself between the medic and his assailant to effectively create a wall for 6 seconds while the heavy / soldier / whatever has time to refocus and destroy them.

Moral of the story: Medics can win arenas. Treat them well.