I often see lamentations about people not being friendly to strangers in big cities. I've never quite understood it. It's one thing to not greet or acknowledge a person if you're already having some sort of interaction with them, i.e. holding the door for someone as you leave a store. Of course you should say something or at least smile and node politely, perhaps making brief eye contact, in that case. But, outside of situations similar in nature to that I actually think it's pretty rude to accost strangers in a crowded public place. People are going about their business and you're attempting to force them to give you attention, even if that's not your intent.
Experience has taught me that above a certain threshold of crowdedness the only people who are greeting me want something from me in at least one of the ways you mentioned. Everyone else is out and about because they have something to do. When I go to the middle of nowhere Kansas from time to time things are different. There you wave to passing cars and say "hi" to everyone even if you've never seen them before. The odds that some random person is going to try and sell you a watch at the intersection of Main St. and State Highway Whatever is pretty much zero. And, there aren't a hundred other things attempting to distract people so demanding a brief bit of their attention isn't rude at all.
Still, there are plenty of residential neighborhoods where no one acknowledges each other on the street, and there is no sense of neighborliness. Paradoxically, given this article, I find that the more wealthy a neighborhood, the less likely you are to be greeted by a stranger. The poorer the neighborhood (I would imagine down to a lower bound – I seldom go to the truly poor neighborhoods simply because they are so far from the city center in New York) the more likely that greeting is, especially in a black neighborhood like Bed Stuy or (when I first moved here) Harlem.
This might be okay if it was someone trying to tell you about a cause that they genuinely believe in but the majority of the time it is likely someone just working on a commission for whatever charity or group they are trying to sell you on.
In the UK people asking for money straight out are usually drinkers. Drug users and scammers tend to want more than loose change hence the more elaborate stories.
I should add that I have been homeless and without money. One time my only option of eating was stealing food or asking for money. I chose not to eat.
I could walk you round a city in the UK and I could point out the real homeless and desperate people that you would never see - mainly because these are not the people asking for money.
I appreciate there is a huge difference here between UK and US since we have a welfare system that is many times more generous than yours. The people you encounter may indeed need the money?
Which was my point really. Of course there is no shame in asking for help but not everyone does. It's partly why I ended up homeless.
I applaud your generosity though.