In theory you are right about them wanting to pass it right away.
But I just can't see it happening right away in practicality, because:
A- they would need 60 votes, which is not a sure thing at the moment. And if they go with less, they need to do a budget reconciliation which is limited to 1 (or 3) per year, so it would take time, especially if they want to tie it to something else in a budget reconciliation.
B- they need the IRS to help with distribution, and the IRS needs to open up the 2020 tax filing season later this month. They simply couldn't handle another new distribution, especially when any new bill will surely have new rules, like older dependents..
Therefore, I see Congress saying, go file your 2020 taxes, go get the Eic money, and after April (when the IRS quiets down), we will work on $2000.
But this is all logic, and not always do things work with logic...