As a mainly crit rider, I think aero wheels help a lot in most crits. The ones with super slow hairpin turns (say, under 15 mph) are the exception. And this doesn't even apply to breakaway/chase efforts where aero definitely matters.
If you're in any sort of a normal crit, you'll be going pretty fast most of the time. It's unusual to slow down a lot for turns, and if you ride smart, you don't have to actually sprint out of turns too much - and that's no matter where you are in the field, not just the top 10 or whatever. You'll probably cruise at 22-32 mph (turns and straights included) and spike up to 40-42 mph (mainly flat course).
The aero wheels really help when the speed goes through the roof - a 5-10 mph surge for example, or you need to move up. If the field is already riding over the 30 mph range, you'll need to make a 35-40+ mph effort to move up. A box section wheel is hard pressed to go over 42 mph (just from experience, no backing data). However, it's not unusual to be able to move up at 44-46+ mph with aero wheels if you have a slight wind/draft or hill assist. Such a spike in speed is very difficult to make with box section wheels.
Aero wheels are especially helpful when the field is strung out or there's a crosswind (esp those wheels that do well in a crosswind). Once the field is strung out, the draft is not as significant as when the field is bunched up. You're going to have to work to keep your spot and an aero wheel will help you do that. I specifically bring my Specialized TriSpokes (HED3) for windy, flat airfield crits. I used to run a rear disk as well.
Finally, in a sprint, aero wheels could potentially increase your top speed by as much as 6 mph (personal experience). It seems that many riders have about the same wattage output in a sprint (i.e. 1500 watts peak seems sort of normal). Even if you jump at the optimal time, your 1500 watts only gets you a given amount of speed (frontal area, slope, wind, etc). If you can get even 2 mph by using more aero wheels, then you should.
I don't know for sure but I think the whole deep rim thing (and disks) have to do with limiting the speed the *spokes* hit the air. This is because the spokes are by far the fastest moving object on the bike. If you can reduce the spokes' drag, you'll reduce the bike's overall drag significantly since the rest of the bike is going at a much slower speed aerodynamically.
If your bike is going 30 mph, your tire is stationary on the ground (unless you're skidding). The top of your tire is going 60 mph. You can imagine your wheel sectioned so that the hub is 30 mph, 1/3 down is 20 mph, 2/3 down is 10 mph, all the way down is 0 mph. Then you go the other way - 1/3 up from the hub is 40 mph, 2/3 is 50 mph, top is 60 mph.
Based on a standard wheel (with tire) height, going 30 mph, the top of the spokes for the following shape rims hit the wind at the following speeds:
box (9mm) = 57.5 mph
slight aero (22mm height) = 56.25 mph
50 mm rim = 53.8 mph
66 mm rim = 52.4 mph
There's various other factors like rim surface area, the way air flows, etc, but nothing can dispute the fact that if you have a 700c wheel and you're going 30 mph, the spoke (or perhaps the spoke nipples) will be traveling at the speeds above based on rim depth and a 19 mm tall tire (I used 673 mm for overall wheel and tire height).
The spokes on the very deep section rim are hitting the air 5 mph slower compared to the box section rim. If you can go 30 mph on the box section rim (turn the wheels so the spokes hit the wind at 57.5 mph) and you switch to 66 mm deep rims, you'll be able to do about 33 mph for the same effort. Nothing except swapping wheels. Keep in mind this doesn't account for your body's drag - but it's a good way to compare spoke speeds to see what potential gains you might achieve by using different wheels.
On 50 mm rims, you'll be able to go 32 mph. Again, no additional effort. Same position, just gear up a bit for the higher speed.
On 22 mm rims, you'll go about 1/2 mph faster, about 30.5 mph.
I have a spreadsheet which does the calculations based on overall wheel height, bike speed, and rim heights you specify if anyone is interested.
The only time aero wheels don't work for you is if you have a very "jumpy" course - things like hairpins, wide straights dumping into a narrow and steep uphill, or a finish line less than the required 200 m from the last turn. Then accelerating to a very high speed in 1-2 less pedal strokes (from personal experience - 4 or 5 pedals strokes instead of 6 or 7) would be better than being able to accelerate to a higher top speed (which would take a total of perhaps 8-10 pedal strokes).
As for handling, aero wheels are definitely usable in crits with winds up to 30+ mph. You have to train on your front wheel so you get used to steering the wheel versus steering your hips - aero wheels get steered like a rudder. Once you do that you'll be comfy with aero wheels virtually all the time.
My helmet cam vids have me running either a DV46 (46 mm rim) or a Specialized TriSpoke in the front. You can see that my handling isn't affected by the wheels (and some of the races were quite windy).
For those that race in CT, NY, NJ, MA, and RI crits, this advice does not apply to you
cdr