They need to be at least trainable to an adequate level, sure. They don't want to debut a member who can't keep up with the choreo. But the fact is, Big 3 can afford to take only the creme de la creme. Yet many Big 3 groups have "that" member.
People used to make threads about "Why is X in the group when they can hardly sing and Y has the same role and does it better, and they are hardly given anything to do?" These days you'd immediately get the appreciation thread of the group running in to rip your head off, but the fact is there are some debuted Big 3 idols who aren't talented and don't have the X factor. They're just there. I'd list some names, but I don't want to be bombarded with YouTube compilations proving me "wrong" all day.
But on the other hand it would also be a big risk for a small company to take on rich kids just because they're rich. If they're not bringing any other assets it's very unlikely their group will amount to anything, other than appearing in the occasional "under-rated girl groups" compilation YouTube video.
Too many inaccuracies and assumptions in your post. If big companies don't need the funds and connections their trainees can provide, it means that rich trainees getting into big companies is a matter of coincidence.
I can also name many idols from small companies that aren't particularly talented but still debuted anyway.
Well but even with applying all those it ain't easy to get fame/level/status of BIG3... This is why no one is using BIG4 even when BigHit is doing better than all BIG3 companies... Look at RBW or Cube, both have amazing artist yet no one would say that there is any privilege when debuting in those companies. Yeah it's better than random small/mid company but still it don't guarantee you high sales or 100 mln youtube views, as well as it doesn't guarante you entering charts.
Hype level is hella different so in this case SM, JYP and YG privilege will always exist.
Debuting in the big3 does not guarantee high sales, views and charting either and it also brings a different kind of burden other companies groups don't have to worry about.