Members of the New York City Council united on Tuesday behind proposed rules that would democratize the bill-drafting process and the way in which discretionary funds are apportioned to individual lawmakers — steps that fell short of Mayor Bill de Blasio’s call to do away with such spending entirely.

The proposals were billed as a corrective measure after the eight-year tenure of former Speaker Christine C. Quinn, during which two council members were indicted on charges of misusing discretionary funds.

The proposals were championed by her successor, Melissa Mark-Viverito, a Democrat from Manhattan, though they would strip the speaker of some of the powers of the post, prompting several council members at a news conference on Tuesday to praise her for what they called an unselfish act.

“To me, power is in being able to have a legislative body that is collaborative, that is engaging in having a vision for New York City, that is going to be able to do that with minimal obstruction,” Ms. Mark-Viverito said.