First off, without incentive to drag oneself off of the "given enough to live" level, you do end up with problems already mentioned about people happy to 'stay where they are'. From a UK perspective, there are the 'chavs' and 'sink estates' and "Shameless" mentality (ref: UK TV show) and a story in today's paper about a mother of six who has somehow (this article is short on detail) found herself emplaced in a £2million house in London, with rent of £1,600/week paid for by the Housing Benefit scheme. Although admitedly the OP's original premise looks like it would at least work against this latter case and stop such strangeness.
Government funded education isn't charity, its a investment. Having a educated workforce to export skilled labour and import unskilled labour is what separates economic super powers from developing countries. The only reason the United states can have such a poor education system is by being a strong enough economy to attract skills from all over the world.
My reply to this point, though, which is why I'm sticking my nose in, is that Government-mandated education beyond secondary level (i.e. Higher and Further education: which is Colleges and Unis in the UK although I know the USA "College" isn't the same as ours) has problems like saying "We're short of engineers! Give incentives for students wanting to qualify as engineers!" then flooding the system a few years later with a whole lot of engineering 'graduates' of vastly varying ability but all with pieces of paper, meaning that the engineers we need are smothered by a whole lot of others who were good enough to pass the exams but haven't the same calling. And about this time someone decided that there's a shortage of nurses, so implements incentives for nursing which similarly backfire at about the time someone else tries to correct a lack of teachers by the same method... And each fresh new government (or just before each set of parliamentry or local elections) swaps and changes it how it feels like, having used the glut to bash their predecesors.
On the other hand, with a completely free hand, you end up with something like the oft-ridiculed "Media Studies" glut of recent years. (Though in my day, it was Art Students who were the supposed 'free wheelers' through education, and some of the 'soft sciences' fulfill both this and the "we need <foo>!" mentality mentioned above.) Today, I'm sure "doin' somefing wiv compooters" is an attractive option. And though the latter, especially, might cause me problems as they're invading my chosen field and reducing my own opportunities, I must clarify that I'm not suggesting that the fields themselves, or even a majority of the students in them, are 'easy option, back-of-the-cereal-box' in any way, just a distillation of the perceptions that can be made (and often spouted by everyone from left-wing comedians to right wing media).
Also, insert whatever job-types you feel like in there, though. Engineers/Nurses/Teachers was just an example, and probably in no way connected to reality.
So. Solutions? I'd go down the road of a hybrid Apprenticeship scheme. Make it so that companies can (with appropriate support and guarantees, especially in the case of smaller businesses) commit to taking on an unqualified person direct from leaving compulsory education (who passes some appropriate aptitude tests, and expresses a genuine interest in the role) and allow company and apprentice to 'invest' time in each other. And it should be open to all businesses, from silversmithing to funereal services, from architecture to... well, computer support.
I suppose you could tweak it a little to try to encourage a little bit of new-blood (hopefully not literealy, remembering H&S at all times) into industries such as farming, where the average ages severely need to rebalanced by a slew of keen youngsters, but supply and demand (though not in a capitalistic manner) ought to be able to help that balanced[1] without forcing unwilling youngsters into fields, or forcing the adoption of unsuitable candidates from the company's POV. If I'm not sounding a little Utopian in my all-too-briefly sketched ideas.
[1] As a caveat, I do acknowledge that if an industry (like farming or 'Little Mester'-style metalworking workshops) gets below a certain level of existing presence, this reduces the number of potential placements to below the usually viable degree, even if you allow doubling-up on apprenticeships and can somehow support that. I have a few ideas about what to do about that (e.g. the way that stone-masonry and other largely extinct arts are 'rediscovered' and reinvigorated by medieval cathedrals who gather together what experts there are in the field and build a team to re-learn and implement anew the old skills in an authentic manner, with both practical artisans and academic eyes involved), but they would need to be finessed a little to be actually workable. OTOH, at the top-end, apprenticeship opportunities in some bloated field (e.g. call centres) would be proportionately less than the current workforce, and might even be exempt/overlooked given the nature of the in-the-job-training that's already carried out.