(why work if you can get money for free?)
This is a linchpin of your entire argument, and I do not believe it stands up to basic scrutiny.
First of all, I don't think anyone's proposing a basic income that would put someone who does not work for a living at a comfortable middle-class lifestyle. Particularly in its early stages, I would expect such a provision to net you about what you'd get working full-time for minimum wage—which, right now, is somewhere between $15k and $20k per year.
I dunno about you, but if that were my "basic income," I'd still feel a need to work for a living. It would be a huge relief to know that I had that safety net—that if I lost my job, I'd still have that much guaranteed to me—but I would have no desire to rely upon it as my sole source of income.
Second of all, even if the basic income amount were enough for you to live at a level you were content with (and note that that would have to include any discretionary spending you wanted to indulge in, like travel), I know a lot of people who would just never be happy without some kind of meaningful work to do. Sitting at home doing housework, watching TV, or surfing the web would get old for some of them within a month or two, for others no more than a few days.
Third, one thing that prevents a lot of people from getting work is the fact that they don't have enough money to, for instance, own a car to commute in. Basic income would go a long way to ending homelessness, and allow people who want to get jobs, but can't get together enough money to look presentable for a job interview, or even travel to a job interview, to do so.
Beyond these basic points, it's also important to consider the ways in which basic income would change the shape of employment. Liquidity in the labor market would skyrocket, for one thing. If the consequence of quitting your job because you hate it, or standing up to an abusive or negligent employer, is no longer "homeless within 6 months, dead within a year", a lot more people are going to be willing to do that. This shifts the balance of power hugely away from employers and toward employees, compared to where it is now—especially when you consider that there will, in all likelihood, be a fair number of people who do voluntarily leave the workforce entirely to live on basic income. Furthermore, part-time work starts to look significantly better when you don't really need the money that working an extra hour or three a day gets you. So not only are you much more likely to get a job that you actually like (assuming you're bright enough to have gained the skills to do such a job), you get to have more leisure time to do the other things that you really enjoy. And if you don't care about making most people's lives better...then just consider that there are probably a dozen other ways in which the fundamental shape of things will be changed by implementing a meaningful basic income, so assuming that it would be impossible to pay for because "no one would work if they got paid for living" is just lazy and unsupportable.
In the end, it's quite possible that basic income would provide a net boost to the number of people employed, and nearly certain that it would provide a net boost to productivity.