I'm saying your narrative that 2nd gen groups could tour just like the top 3rd gen groups is inaccurate. SNSD could not tour outside of Asia. They made one TV appearance in the US and were never heard from again outside of KCON, which isn't even an individual concert.
What BP (and others) are achieving is impressive for a Kpop group. There have been bigger girl groups than them worldwide however. They simply don't have that sort of competition to worry about these days, so it's easy to say "biggest girl group in da world!" Same goes for the 2nd gen groups. No one else was putting out popular girl groups at that time.
Legitimacy is important because... when isn't it? Having people worldwide know your name is objectively better than having only Koreans or Japanese know who you are. It doesn't matter if you can fill the Tokyo Dome, this will still hold true. And guess what, Twice even filled the Tokyo Dome recently! Three times! But the fact that they're popular enough to also tour in places where previous groups could not is, yes, more impressive! I'm not downplaying anything, it's common sense really.
Ok I zero clue what you're point is anymore.
So you just said even BP touring achievements have no merit because they can't definitively be the biggest because they don't have competition. That doesn't make a lick of sense.
And yeah you are downplaying Asia. Saying Tokyo Dome doesn't matter (lol) when it's a huge venue both physically and status wise.