That's not true, at all. Congress has power to WRITE bills. The president still has to sign them. It always takes a 2/3 vote to override his veto.
The original process was for the House to have to take positive action to fund the government, with the Senate and Exec having some say.
What has happened is that various methods are used to fund the government perpetually, with positive action being required to halt funding.
It certainly *was* intended that the House could simply refuse to fund areas of the government, with the Senate, Exec and interested private parties left to be mere supplicants.