Why? Isn't it easier to pick the fathers more carefully at the beginning?

Wow... that's an understatement I picked my children father I repeat father meaning one very carefully married for 25 years but nothing last forever with that being said fathers on SSI should be reliable to pay child support...

Well, as far as the SSI... there are reasons what that is considered untouchable money. Good reasons. If you are on SSI... its because you CAN'T work... that means you can't get a second job to cover extra expenses etc...

And on SSI... you aren't swimming in money. Its hard to live on it even without extravagances or debts, and generally its not by choice, but by need.

Not making any excuses for deadbeat dads... but there is a point that extracting that last bit of money is no different from Shylock in Shakesppeares The Merchant of Venice... trying to get his pound of flesh.

Which if you have never read it... please do. You should be able to Google the entire play up.

First yes, it is sad that children in those cases are often the ones forgotten, but as noted, the income they get on SSI is so low that in general there is no income

Perhaps change laws so that The government would have to pay a percent of the SSI directly to dependent children, but that is your question, how.

Get enough elected officials who want to do it, and get enough of the population willing to raise taxes on someone or something to pay it.

How can the law be changed so that fathers on ssi pay child support.

The same way ANY law gets changed. You contact your congressman and senator and implore them to sponsor a bill to change the law. You will probably need to get a significant number of their constituents involved to get them to take action. Then you will need to get a groundswell of public opinion to convince other legislators to support the bill.