Probably at least once a week, you're going to see someone drop something, press the wrong button on the elevator, try to push on the pull door etc. My own stress always peeks when I'm in public and trip up on something minor like that. If you just shout "hey you need help?" you're probably not going to make them feel any better and doubly worse if you just avoid eye contact and walk around them. When that happens to me, it can reaaaally sour a morning.
But saying "they really need to make these doors automatic, I dropped my coffee here last week!" and helping out if they want it has the EXACT opposite effect. Suddenly, it's not embarrassing any more, and you might have a little convo commiserating about what sucks. It's just a little bit of connection to make someone's day a bit better which is definitely a win-win and good way of breaking the seal on talking to folks.
Maybe it's age related, but if any of the scenarios you wrote happened to me, I would not be embarrassed receiving someone's assistance.
Coming up with the right phrase in the moment sounds risky. Is there another phrase you can use to offer help ?
> If you just shout "hey you need help?" ...
There's also another more neutral option. Just give them the answer they need without the fluff. If they then want to thank you and chat it's their choice, but completely optional.
This is probably not the right approach most of the time, but it works well on the types of people who seem "serious" (not anxious or upset).
This is a really hard one to pull off. You have to determine that they really are that type of person and then just magically know what they want. It's really satisfying when it works though.
I've met some of the most interesting people I've ever known that way.