WheresWaldo's post has it exactly right. It's hard to argue with the idea that starting off two feet (one captain foot and one stoker foot gives both riders more control over their own fate) isn't safer than starting off one captain foot. Safer stopping too. Using sticky-soled MTB or touring shoes is safer than slick plastic soled racing shoes because the MTB shoes are more likely to stick instead of skidding.
Once you've committed to starting off one captain foot and one stoker foot, leaning the tandem like a single bike makes a lot of sense. Us tall guys tend to forget that the 10.375-10.75" bottom bracket height standard on tandems is a much greater fraction of a stoker's inseam. You may HAVE to lean the bike so the stoker isn't sitting on top of sensitive parts being squished by the top tube. You steer under the lean to the degree necessary at the start. Everything (pedals, saddle, bars) is proportionally higher for a stoker because of the 10.75" BB height.
Timing a start is better than counting on it to work. Starting in a medium gear (too low gives insufficient speed, too high is too hard to start) you (the captain) say, "1,2, 3, go," and give a half stroke to the bottom of the pedal circle with the strong legs (both on one side, of course) that were already clipped in. That brings the other pedals up to the top of the circle, where you both clip in while coasting. "I'm in, you're in (in the form of a question or a statement, depending on certainty)," and then "go."
For a stop, the captain will say "3,2,1, stop," while applying the brakes. On some given number, usually
"1", you both unclip. A slight steer away from the unclipped feet ensures that you will gently tip that way, and on "stop" the bike stops and the feet touch the ground.
I have seen a few people drop their all-the-way-clipped-in stokers over the years. Sometimes slippery-soled racing shoes were implicated as well. They often work if the foot goes straight down, but with a little bit of angle, they will slide right out from under you. It's a long way down for a small stoker. I have yet to see a two-foot-down team fall over, though to be fair I have seen a few starts by beginning teams that required a stop and a restart, owing to problems with coordination between the members.