There’s a character who is an arrogant boxing champion in the anime/manga “Hajime no Ippo” named Takamura who refers to himself as “ore-sama” which is something like “Mr. Me”, but turn the arrogance up to 11 and that’s the connotation…it’s quite hilarious but there isn’t a good english parallel I can think of.
You’d have to read the song (it is the one I linked to, though it is not a good transcription or translation). Their lyrics are quite impenetrable overall.
Annnnnd, hijack over.
“Mina-san”, which means: "Coming from this humble me to All of You as-politely-I-ever-am-able-to-do-in-this-case-Please-Ok? (regardless of the number of people at present)…
…will do quite fine in (even ut)most (common) cases of the first contact, especially when your body language is clearly showing out that you are now being as polite as you can ever be, but you are still not “crawling”. (Do not crawl. Ever.)
Mina-san is therefore an all good fine Start. What will follow next, well: Lingo skillz. What did you think? 
Off topic.
The following is all mine. Just leave it alone if you do not like it.
The thing with speaking in Japanese while finding your way out is that you can’t always really be that much “assertive” on the spot, not really. I does not work, not always.
The discussion (talk, dialogue, please-get-me-out-of-here-please -sobbing) takes it’s time. There will occur certain pauses being actually not intimidating at all, it just means there will occasionally appear this certain “space” in speech. Giving and taking “space” is polite.
It is (in a way unfortunately) also a mindset, however you can indeed synthesize a mindset for yourself for this specific purpose, and it is all good.
However again, it is much better to lose yourself in translation than in a mindset. Trust me on this.
Well, my 2c.
Yours,
Sasamu
I think the only English parallel is “Please punch me in the face, I think I’m better than you.”
I had to look into it and whadda you know, it could be comprehended pretty much to be so. There is certainly a natural difference between fantasy and real world but surprisingly, the answer would be yes.
Offtopic again (really excuse me for being an american, european and asian at the same time, or, whenever it takes)
When being On Purposely arrogant, you are saved or lost in very different kind of ways regarding situations, for example when comparing Japan and USA, or Germany and (every) Nordic nation.
I honestly think that the “Ability to Read the Mood” as a subject should be included in every curriculum around the world. Ah, well, no need in such case if you have both guns and money enough.
Yours,
Sasamu
“Beg pardon, but that’s MISTER ‘The Top Man’ to you, peon.”
I’m sorry, but it really doesn’t mean this. Even “Mina-sama” doesn’t mean this.
Yeah, isn’t mina-san basically just a minimally polite “y’all”?
Well, unless you’re using 御名.
OP referred to a situation where he/she needs to introduce him/herself to a bunch of previously unknown people.
If it is not about persons you Really should know in advance before formal negotiation (like the boss or other significant superior you need to confront in a positive manner) then mina-san is quite adequate way to acknowledge the formerly unknown people around you.
If the OP picked up a girlfriend/boyfriend and it suddenly got serious, there are then more defined expressions to deal with the situation.
It’s not just about keigo, it is really also how you express yourself. And Do not Crawl in front of Anyone. Ever. I know this by experience.
It is certainly very different situation when writing a formal letter to a potential employer or official. In such case you need a native writer you can trust.
Yours,
Sasamu
OP said AN unfamiliar person, as in singular, where “mina-san” would be inappropriate since it is AFAIK intrinsically plural.
Yeah that’s the inferrence. 
Ah, my bad being illiterate, indeed.
Yours,
Sasamu