My personal thought is, who cares? You get what you pay for, right? Services like Khan Academy are great if they're helping people learn things they wouldn't otherwise take an interest in learning about, or if it enables learning they were interested in but couldn't afford traditional methods of education.
If you're already IN a traditional classroom environment, then no - I'm not sure Khan Academy lessons are so great. I mean, you have to ask, as a paying student, why you're paying your hard-earned money to get a personal classroom experience with supposed educational professionals, who turn around and ask you to sit through canned Khan presentations instead of presenting the material themselves.
As for the "precise explanation of mathematical concepts to be mere nitpicking"? Maybe it is, really? By that, I mean, most people are really only interested in learning math as long as it allows them to accomplish something. The minority who find the theory itself fascinating and want to learn more math for the sake of learning it are the ones who will probably move beyond whatever Khan Academy teaches, and consult other sources.
If you know enough math to get correct answers to the problem you encounter as part of your daily life or job, then that's likely ALL the math you really need to know.
You never get what you pay for when the government is paying the tab with your money. In the US, the student loan program has priced a decent education out of the reach of all but the very wealthy and put millions of others into debt hell. You would need a masters degree to get a job that pays enough to work your way through college.