It sounds like you may be pushing off with your shoe (?). If this is the case you'll have a hard time accelerating. Push off with the foot that's clipped in, you'll be across the intersection before you know it.
You can pedal across an intersection without clipping in, if necessary, even one that is uphill. Maybe not San Francisco uphill, but most normal uphills.
If you cannot clip in relatively consistently (8 out of 10 times) without looking down you're probably not practiced enough. I'd recommend doing the following:
- lean up against a wall or similar
- clip in
- see where your shoe is relative to pedal (i.e. "my strap is right next to the axle")
- clip out, back pedal a revolution, clip in again.
- do it 20 times in a row without messing up. If you mess up, i.e. shoe doesn't clip in on first try, then reset your counter.
- go out and do it 20 times again but this time tooling around your neighborhood. Same deal with counting, if you mess up, do it over again.
- you can do the first 40 in a row while looking. Based on the foot push off method of starting, you'll have about 5-10 feet of coasting on one downstroke. This should be plenty to get clipped in if you are looking down. It should take you maybe 1/2 second, especially if you've done it 20 times in a row without messing up. If you haven't done 20 in a row, go do it.
- once you can do it while looking down, it's relatively straight forward to do it while looking elsewhere (like at cars, lights, people, etc).
- to practice unclip and put a foot down at every stop sign, every red light, etc. You look like a respectable cyclist and you practice your technique, a win-win.
In 2 days you will be fluently clipping in.
cdr