He cites "the Census Bureau" for both the numbers and the analysis, but I'm a poor skeptic and cannot find his sources. Does anyone else know where this came from?
The US Census Bureau website provide data on poverty definitions and number of people in poverty in the United States.
See Income, Poverty and Health Insurance Coverage in the United States: 2014
The nation’s official poverty rate in 2014 was 14.8 percent, which means there were 46.7 million people in poverty.
The poverty thresholds by year are available at Poverty Thresholds
The threshold is a function of family size, number of children, and whether adults are over or under 65. For example for 2015, a single individual under 65 is in poverty if income is less than $11,367. A family of three is in poverty if income for the entire family is below $19,078 if it is a family of two adults and one child. In all cases, the $17,000 per person mentioned in the OP would significantly exceed the technical poverty line. More particularly, $12,331 per person is the maximum value in the poverty threshold table.
The remainder of the question is somewhat a matter of opinion. The trillion dollar value comes from reports like The American Welfare State: How We Spend Nearly $1 Trillion a Year Fighting Poverty—And Fail and Examining the Means-tested Welfare State: 79 Programs and $927 Billion in Annual Spending, but it depends how you define "welfare programs".
A trillion dollars is enough to give 46.7 million people $21,000 each.
However, that does not necessarily mean poverty would be eliminated, even by the technical definition of poverty. One would need to consider how the individuals would spend the trillion dollars versus how it is spent currently, as well as how many people might stop working if the money is handed out as proposed.
The trillion dollars includes the Earned Income Tax Credit, which is already giving lower-income people money, but on the condition that they work. Just handing out the money without requiring work would obviously reduce the incentive to work.