Your use of the example "going to school and improving future income potential" is a very externality caused by intervention, i.e. the modern public schooling system and licensing requirements. A more voluntary society where autodidacticism, guilds, apprenticeships and decentralized democratic schooling are practical options for advancement (that a hypothetical laissez-faire arrangement would make easier to emerge) would therefore mitigate the effects of time preference, in that self-advancement would have less legislative barriers and the opportunity for market signaling would be more unfettered now that the principal-agent inefficiencies caused by formal credentialism are lowered. That is to say, employers would rely more on real signals rather than the nominal signals emitted from credentials and licensing that do not account for time and skills corrected after their acquirement, and don't suffer from disequilibrium caused by poorly thought out credential requirements.

Why would we see people expecting to train children in tasks when even today you see adults who need only a fraction of the training a child would not being trained on the job as hiring managers look for already qualified candidates?

Even if this is what society would revert to once it ran out of already trained individuals, it would be cyclic and would cause entire generations to be lost as it would always be preferable to hire some adult who has basic training over a child who needs to be built from the ground up in basics.

Consider the simple case of a programmer. Who would hire a child who would take years to train (as they would need to learn not only programming, but math and reading and writing, among other fields of knowledge)? Even if the child would work for free for those years of training (which would be problematic for other reasons, but assuming they would anyways), it is still not worth it because the cost of training other than salary would make hiring no one a far better choice.

It costs a lot to train a child from scratch and they offer no benefits except in the longest of terms, so people would always hire from elsewhere, such as children who had parents rich enough to afford them basic education costs out of pocket.

Apprenticeship would work well in today's world if you start after the basic education has been completed before specialized education begins, so that it would be quick enough to turn apprentices into employees who could produce value. But this would be later high school or early college; maybe earlier for simpler trades. It will not (and historically did not) work well for younger children with more educationally demanding jobs.