I have been a director for years.
Here's what I want:
1. a few people who you give permission to pick up your child (in DC licensing lingo it's "receive child at end of day."). This means if you can't get there for whatever reason, I need a few people who you can call (or I can call if necessary) to come get your child. I cannot just hand a child off to just anyone, I need permission. So a neighbor, a co-worker (if you are at an on-site childcare or close to work childcare), and a parent or two from your center/preschool will all do. (also, if you just want your child to be able to go home after school to have a playdate in the afternoon, I need permission for that parent to take your child)
2. make medical decisions should I not be able to reach you or spouse (if you have one) in an emergency. In an emergency, a hospital will save a child's life but will not set a broken arm with just my permission. They will wait until they can reach... someone. So list a grandparent or aunt or two, no matter where they live, because they can give approval from California over the phone. Or your child will have to wait until you can get there.
Why have I needed these?
Examples: your car breaks down on your way to pickup and your spouse isn't available (due to travel, being a single parent, being in jury duty and they can't be reached, your spouse is in hospital, etc) In this case, I want someone to take your child home at end of day.
Another example is when both parents think the OTHER is picking up - so nobody gets to the school. A neighbor might be able to get there sooner than you can given you are still at work. Your goal is to get your child out of the school soonest. (don't judge, this happens to many parents, even the most on-the-ball, organized ones - it happens once a year at our program, on average, just not to the same parent twice)
You are traveling and your spouse is notoriously difficult to reach (some govt jobs don't allow you to keep your personal phone on you and don't allow the fed # to be given out), or because your spouse is always in court and can't have phone on, or..... whatever. So if you are the "parent easiest to reach" and this one time you are in an airplane I can't reach you. So then I would need EITHER the emergency # (If I'm heading to hospital with your child via ambulance) OR the pickup people if nobody comes to get your child.
Really, I don't tend to use those #s but I have a few times over my career.
Also, when we NEEDED to reach parents in an emergency and couldn't, we called those numbers to see if they had good ways to reach the parent - a neighbor could knock on the door, a grandparent knew the other cell phone #, etc.
And, yes, if you can have your phone on vibrate in meetings, that's best if you don't have a receptionist.
Also, I LOVE secretaries /receptionists because when I say "this is the director calling from her child's school and I need to speak with her now" they will move heaven and earth and race around to find you! They will pull you out of the bathroom, any meeting, they are great! I don't do that for just anything, but if it's an emergency then I use that card. Otherwise, I will be careful to say "this is not an emergency, just have them call me."