The TSAB doesn't kidnap. It doesn't 'grab' mages. They just offer people with magical talent opportunity to sign up if they involve themselves in one of their operations (which often are to contain and seal lost logia that may have gotten to a magic naive world).
You are correct, the TSAB doesn't kidnap mages, and gives an offer only to mages who accidentally found themselves in the middle of Lost Logia crysis. Trouble is, these mages are the only ones that actually matters. If a stable magical tradition wasn't established in a thousand of years, chance it will appear without outside interference, are miniscule. By offering a job to any mage that found out about them, the TSAB nullifies consequences of such an interference, and ensure that non-magical worlds remain non-magical.
Of course, mages in question can refuse, but, as
@Hate Fish said, it's difficult to refuse, because the TSAB offers much more than a non-magical world capable to. Actually, a non-magical world cannot offer anything, because they have no idea what's going on!
And, I'm quite sure, if the mage asks why the world cannot be informed about magic and the TSAB, the TSAB will truthfully answer that it's a horrible idea. It will cause chaos, governments will start new arms race, with the mage in question as the focal point of it. Whatever happens next, it will surely destroy their life and hurt hundreds of other people.
So, it looks like to take the TSAB's offer is the most beneficial for all sides.
The mage will have a chance to develop and use their powers.
The TSAB will receive additional asset to deal with their many problems.
The non-magical world will avoid chaos and uncertainties of the First Contact and/or the Magical Revolution. But, of course, it's only beneficial in short-term.
Because magic is incredibly useful tool, capable to drastically improve quality of life.
Because magic is the only way to fight magical threats.
And, while revealing of magic will certainly be hurtful, every major development in humanity's history was hurtful.
The Industrial Revolution hurted thousands of people, the atrocious Soviet Industrialisation hurted millions, but I, for one, glad they happened.
Though frankly, given how Earth Bet is... I kinda doubt magic will help make it better. More than likely the various governments and groups will commit to horrific experiments to try to make mages or something.
It surely wouldn't hurt.
Parahumans are the main problem of Earth-Bet. Even with Zion gone, even after Endbringers will be defeated, triggers continue to happen. Damaged people, additionally conditioned to provide maximal conflict, continue to receive incredible powers and use them for their own advantage, which usually means crime.
The result is obvious: the USA, probably the most well-off and stable country in the world, have triple more villains than heroes, and even that only thanks to Cauldron's meddling. Meanwhile half of the world already in a state of parahuman feudalism. Without Cauldron this process will continue until Earth-Bet civilization ends.
Introduction of magic let us reverse this process. Unlike parahumans, mages don't necessary have psychological traumas, they need to be trained and need to have a Device to be actually effective.
As long as we control magical knowledge and Device construction, we can guarantee our mages will bolster law-enforcement and not criminals.
Of course, other countries won't be happy with the US strengthening, but people who can start 'horrific experiments to try to make mages', are already doing such an experiments to make parahumans, so... *shrug*
Shas, why is Japan a relevant example?
Because it's the closest example of what I want to do with Earth-Bet.
Not uplift, but modernisation.
