There will probably be some cross-checking, but there will also probably be some errors made, as well.
The vast, vast majority of Social Security recipients do not have income levels that would cause a reduction in the amount of the stimulus payment. The IRS knows who gets social security payments, and also knows the incomes for each person who does. If I were at the IRS doing this, I'd send the Social Security Administration a list of those recipients whose income is too high to get the full amount, along with a calculation of the amount to be paid. Everyone else gets the full amount.
Will they do that? Maybe. I suspect the database systems for the SSA and the IRS are not compatible, so it could be difficult to link the data, but I don't know for sure.
Still, even if they sent the full amount to every Social Security beneficiary without checking income, it wouldn't make a huge difference in the total amount sent, and excesses could be calculated and recovered in the 2020 IRS tax year. All that would take would be one line in the tax forms for listing the amount received in the stimulus payment.
What I do know, though, is that the SSA has the system in place to issue those payments efficiently and in a timely way. So, I imagine that's exactly what it will do. That's the easiest way to pay SS recipients by far.
One of the problems with going to the IRS web form could be duplication of records, which could actually delay an individual's payment or result in duplicated payments, which would have to be clawed back. That's one reason I'm not going to the IRS form right away.