As long as I can tell the parent is trying, I don't get nearly annoyed as I would if the parent was ignoring the child and letting me have to deal with the consequences. It may not be what will work in the long run, but don't let strangers suffer because your (general) child is having a tantrum.
Oh, good. I thought I was going to be the only one in this camp. I think it's fine to use whatever parenting technique works best for you when your child isn't making life unpleasant for others. When the kid's behavior has a negative impact on other people, however, it seems like the best strategy is to remove yourself (and the kid) from the area or at least make every effort to get the kid to be quiet.
I - along with a store full of other people - was recently treated to a jet-engine-decibel-level screaming fit courtesy of a small child whose mother dawdled along the aisles at a leisurely pace, making no effort to leave the store, get out of the place in a hurry, or quiet the child. It went on for the entire 15 minutes I was in the store and people were quite obviously supremely annoyed. Some loud, non-ehell-approved comments were made... The mother may have great success with ignoring tantrums, but the rest of us sure couldn't ignore it.
I think this example is very different. In your example the mom has a very valid option of leaving the store. I think everyone agrees a parent should remove rhe child when possible. In the OP, the moms option to remove the child from public would mean she waited somewhere other than the bus stop and then didnt take the bus.
Why should the public suffer from that? The mom could have moved down the street where no one else was standing/waiting and watched for the bus arriving. She could have also at least attempted to quiet her child to let the public know she was trying. That makes a big difference.
We don't really know what the mom had tried before the OP arrived at the bus stop. What she witnessed could have been the very end of a major meltdown that the mom had reduced and felt further acknowlegement would create a spark again.
I truly tried and in most cases succeeded in keeping my kids from creating an uncomfortable environment to others. But I was not ever going to base my parenting decisions on how a random stranger was going to judge me.
It's not about judgement, it's about respecting the space of those around you. It's no different than talking during a movie because there is no respect for the people around you (all yous general).
You're right, we don't know what the mother tried, but she, out of respect for a stranger, should have at least tried something else to keep her daughter quiet to show that she was aware her daughter was being rude and was actively trying to stop it.
I in no way see a public bus stop anywhere near the level of a movie theater or other in door space. I so often have to hear the music blaring from peoples head phones, listening to their cell phone conversations, or having conversations amongst themselves that I find inapporpriate that a young child whining to her mom wouldn't even put a blip to me.
So here is a scenario
DD got up cranky, isn't happy about having to go out so early, and in a general bad mood. Mom and DD arrive at bus stop.
DD whining: Mom, I want to go home.
Mom: We have to go to X.
DD: I don't want to, I'm tired.
Mom: Well, sit down here while we wait.
DD in louder whine: I don't want to sit down there, it's icky.
Mom: It's not icky, it's fine. Just sit down.
DD getting into full blown mode: No, it's dirty and its rough. I'm not sitting there and I won't.
Mom: Fine just stand. (And puts her bags on the bench on both sides.)
DD goes into full pout, OP arrives and indicates she'd like to sit down. Mom moves one set of bags.
DD: But mom, where am I going to sit?
OP: Oh, is this your seat?
DD: But mom, I don't have any place to sit, I want to sit down, mom, let's go, I don't want to be here, come on mom, there's no place for me to sit, I want to go home. MOMMMMMMM!
Mom sighs, gives DD that look and refusses to re-engage.
If she does, it won't accomplish what the OP wants, which is silence instead in my experience it would be just the opposite, an escalation of noise and whining. So why should the mom re-engage and re-enforce for the DD that she can manipulate mom into an argument or give in if they are in public? Just because some random stranger sitting next to her with head phones on is going to think she is rude?