That's not entirely accurate.
Pelosi needed to get the checks out as soon as possible and was only lowballing to get the Republicans to vote on it. Naturally, when Trump signaled for a much higher package, it lead to the second and third stimulus checks as a compromise. The Republican-controlled congress at the time wasn't so keen on doing this. Ever.
They viewed this as an entitlement. While true, Trump's signaling a willingness of a $2K amount did enable Pelosi's move; it was in vain because she erroneously thought that because Trump himself, the Republican's platform of anything pro-Trump, that they might fall in line. They didn't.
https://fortune.com/2020/12/25/us-covid-relief-stimulus-package-checks-pelosi-trump-republicans/