This is not what universal basic income is. Universal basic income isn't money from income tax going to the unemployed - it is a complete rebalancing of the welfare system which leads to every person in the country - regardless of income, employment status, age, gender, any other metric - receiving a set amount of money each year with no requirements of them to receive it. The eight workers wouldn't be supporting the ten, at least not in such a clear cut manner, because they themselves would be receiving the basic income as well. Most of the funding comes from non-income tax sources, such as corporation tax, or in my somewhat pessimistic example of AI replacing too many jobs, an AI tax that remains to be implemented. Naturally in a purely income tax funded system, the issues you mention are completely valid, but that isn't what universal basic income is all about.
Also I think you're overestimating how much people would receive from a Universal Basic Income. As a rough idea, if it were implemented it would not be significantly more than $5000 a year per person at the most, and even that may be an overestimate. Eventually, as more jobs are replaced by AI, the total pool to put towards a UBI would increase, and each person would receive a larger amount.
(plus the governmental agency that continues to grow as it gains more people to "care" for and has even more checks to process).
As I previously alluded to, as UBI is universal, the government actually saves money these checks would otherwise take, as everyone is eligible by default, and remains eligible regardless of what they do.
Without intending to cause offence, this is an absolutely ludicrous viewpoint. First, my opinion is very much that no person should be obliged to work to live just because their parents decided to have a child, but my opinion is not important nor is it shared by most proponents of UBI.
My main issue with the points you just made is about those who are mentally or physically handicapped, but to suggest people should just work in McDonald's for a pitifully low wage and no job progression or security demonstrates, in my opinion, a troublesome lack of empathy.
The unemployed are not just a bunch of lazy people who can't be bothered to work - in many cases, the wage McDonald's would offer is not sufficient, even for the poorest. Before I go on, I live in the UK, not the US, so what I say may not be directly applicable, but I'm taking this page as an example of the pay for a McDonald's crew member. We'll take the maximum listed for a cashier, of $9.91/hour. Assuming 8 hours of work per day, 5 days a week (which you'd be lucky to get), and working 50 weeks per year, gives an annual total of $19820 per year. This is almost $9000 less than the cost of living for a single adult in the US (calculated here). Clearly just getting a job at McDonald's isn't going to solve any issues here.
However, by far and away my biggest issue with what you said is that the mentally and physically handicapped should work in a grocery store. Perhaps you meant this for some and not all, or perhaps you just don't understand how terrible certain handicaps can be. I'd also speculate you've never suffered from depression, though please correct me if I'm wrong. Certain mental handicaps get in the way of your ability to motivate yourself to do anything, certain physical handicaps cause someone to be in pain 24/7 in such a way that even getting out of the house is a great struggle - these are people who should be looked after by society, and those who disagree are, frankly, completely unaware of the daily sufferings of those with such disabilities.
I'm quite busy currently, but will watch the video at a later point. I hope my explanation in this post, though, displays how the situation you describe the woman as being in is not applicable to a discussion of UBI - one where everyone, regardless of any metric, receives a flat rate of income. My personal ideal isn't a universal basic income, but a basic income for those below a certain wage (say $50000-$80000 as a rough estimate) and further supplementation for those with serious disabilities who need more help, but realistically, a universal basic income is easier to implement than a non-universal basic income which would be my ideal.