... Not saying it would be easy, but there has got to be a better way ...
Agreed! But I think that "better way" remains to be created. I don't think the White Council knows how to do it. I think mostly they're too hidebound to even think to
want to.
There's another problem, too: finding/identifying these proto-wizards. Harry didn't notice Molly's talent, developing in his own (relative) back yard. I think there's no real way to detect proto-wizards, in general. The Sight might reveal them, I suppose; and of course physical, skin-on-skin contact. Hardly practical, either one!
Over in another thread, this same idea has come up. The most-obvious solution seems to be to use the Paranet. They are MUCH more widespread, MANY more eyes on the problem & boots on the ground. They doubtless include professional educators, too. Of course, the White Council considers them essentially non-entities (other than an occasional source of warlocks). But Harry knows better.
... Oh when Harry shows up at a Council meeting there are some sitting around in brown robes, but not all that many.
That could simply be a matter of statistics, of demographics. If any group of humans had a lifespan of 200+ years, the population would skew HARD to the older cohorts, with just a very few youngsters relative to the adults. It could also be the aforementioned problem with finding the damned kids before they've gone sithmode, and their apprenticeship would come with a Damocles warranty...