Actually, it’s only forgiven to the borrowers. The federal government will print the hundreds of billions of dollars to give to the schools on the taxpayers’ behalf, so it increases money supply and inflation. And it decreases motivation for future students to honestly take out a loan with an intent to pay it back. And it decreases incentive for schools to lower tuition fees.
The schools have already received the money. The Fed's hold the debt security. This is the same as throwing that security away - it will never be collected, and the money supply will contract, not expand.