How that plays out depends on the person, and since we're talking about random strangers here, it's really quite unpredictable.
In my experience I've come to prefer being completely anonymous in public. The communities where I've been more outgoing and developed a rapport with strangers, it's only become a burden.
I don't want to have to say hi and explain to people I don't have time to talk when I'm going through the grocery store, or just getting work done at a cafe. It's annoying, and depending on the person, they can appear really quite defeated or offended even, by being denied the opportunity to socialize again.
I've also found that relations are in a better place as complete strangers vs. "ice broken" but with denied interactions more than once. The former is a clean slate, the latter is actually in a worse than clean slate form. You've gone from clean slate to actively rejecting the contact after seeing a bit of what's behind the previously closed door.
Being social with strangers is great if you're extroverted and want to prioritize those interactions in your life. Otherwise it can really be a can of worms. Now I only do it when I'm traveling, since I'm confident I won't see those people regularly. Even those interactions can be problematic, people often want to exchange contact information and meet again, and get offended when I refuse.