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.
Certain classes are better against certain other classes.
In general:
Heavy is weak against Spy, Sniper, and Demoman.
Heavy is strong against Scout, Pyro, and Medic.
Heavy is even (more or less) against Soldier.
An Engineer versus Heavy depends entirely on if the Engy has a level 2-3 sentry or not. A mini-sentry or level 1 sentry will almost always see the Heavy win at full health. In this case, the Engineer should flee. If he was building mini-sentries, building them as you flee is an excellent idea, as it will slow the Heavy down to kill it, or make him ignore it to chase you, dealing him damage all the while.
Medic is weak against Scout, Pyro, and Soldier... well, most of the classes in the game, but those three in particular.
Medic is strong against the Spy, simply because a Medic that continually moves will generally outrun or dodge a Spy's backstab.
The thing is... a Scout has the best chance of killing a Medic if he can manage to get behind the Medic, simply because a medic is a 2-shot kill with the standard Scattergun at close range.
Now, the best defense against a Heavy/Medic pair is to work with your own teammates. If you can take out one of the two, the other teammates may be able to take out the other.
If you really do need to combat them alone, Sniper is likely your best bet. As tempting as it is to go after the Medic first, if he keeps moving, he can generally outrun your scope. However, a fully charged headshot will one-shot the Heavy.
If they Uber, run, unless you're an Engineer trying to keep a sentry nest up. In this case, repair repair repair, and hope that they aren't smart enough to have the Medic block the sentry's knockback effect. If you can get a Pyro to help you by airblasting the Medic or Heavy away, even better.
P.S. NEVER go head-to-head versus a Heavy, he will always likely win simply because of how much damage he does.
Best Answer
Here's what you can do to sentries as a scout if no engineer is around:
Here's what you can do to sentries as a scout if their engineer is around: