Columbus, Ohio(CNN) If not for the promise of President Donald Trump's signature, the current Republican effort to shred Obamacare would have ended like so many others over the past seven years -- defeated at the pass.

But this time around, with an approving executive itching to sign their work, Republican leadership is pressing toward a comprehensive overhaul, ignoring pleas from Democrats on Capitol Hill for more open debate, and furiously whipping support from wobbly GOP legislators whose defections could imperil their progress.

Trump, for once, seems to be an afterthought. In conversations with more than two dozen attendees at weekend events headlined by Sen. Bernie Sanders to protest the Republican bill -- first in Pittsburgh, Pennsylvania, on Saturday night and then Columbus, Ohio, early Sunday -- the President's name never came up unprompted.

On the sidewalk outside a small concert venue in its Arena District, Columbus resident Kelly O'Rourke, 55, said the political tab for any potential harm the law might do to a grandson born with health issues or her own costs would come due on Capitol Hill.

"I'm going to blame (Senate Majority Leader) Mitch McConnell and the guys who have been there for a million years," she told CNN, dismissing the President's role.

"Donald Trump is not a politician," O'Rourke said. "He has no idea what he is doing. It's not Donald Trump. It's everybody who's in there, laughing at us, thinking, 'Ha-ha, ho-ho, they have to look at this circus going on and let's -- whisper, whisper, whisper -- go do this without anybody knowing.'"

The Senate Republican leadership is pressing for a vote on their overhaul later this week with the goal of passing the bill before lawmakers leave Washington for the July 4 recess. House Republicans passed a similar bill by a narrow margin in May. If the Senate GOP conference can manage 50 votes from their 52 members, the two chambers would likely hash out the differences, then send the paperwork to the Oval Office. So far, a handful of Republicans have said they oppose the bill in its current form.

Outside Washington, the calculus is more complicated. Democrats, along with advocacy and activist groups, have been appealing to the grass roots to flood the offices of Republican elected officials with calls demanding they disown the deeply contentious legislation.

Saturday night in downtown Pittsburgh, Sanders, the Vermont independent who caucuses with the Democrats, rallied more than 1,500 people against the bill on the first leg of a hastily assembled joint barnstorming tour with MoveOn.org.

Passage of the Republican plan, he said, would be "a moral outrage that this nation will never live down."

"This so-called health care bill passed in the House last month is the most anti-working-class piece legislation passed by the House of Representatives in the modern history of this country," Sanders continued, calling the Senate's version worse and again warning that thousands of Americans are at risk to die for lack of care under its provisions. "It is unconscionable, and it must not be allowed to happen."

Sanders has been criticized for painting the potential repercussions in such stark terms. Still, he argued the same on Sunday in Columbus, citing academic assessments and noting he delivered the message "with pain, with anxiety." During a brief conversation with reporters in an Outback Steakhouse off Interstate 70, on the road from Pennsylvania to Ohio late Saturday night, he quietly cycled back through the argument and, not quite throwing his arms up, said again of the numbers, "It's true!"

The Congressional Budget Office could release an assessment of the effects of the Senate legislation as soon as Monday, a Republican congressional aide told CNN. The office released a report in May saying that under the bill House Republicans passed last month, which has much in common with the Senate proposal, 51 million would be uninsured by 2026, 23 million fewer than would be covered under Obamacare.

The rolling protest, which hit three states in less than 24 hours, shipped off to West Virginia for an afternoon event in Charleston on Sunday, giving Sanders and allies another chance to turn the screws on three Republican senators -- Pat Toomey in Pennsylvania, Rob Portman in Ohio, and Shelley Moore Capito in West Virginia -- whose states are expected to take disproportionate hits from the GOP's Medicaid rollbacks.

In Columbus, Sanders trolled Portman, who has expressed concerns over the Senate GOP legislation that he himself helped craft in a secret Senate working group, with some advice: "If you don't believe what I'm saying, listen to your own governor, John Kasich."

Only a few hours earlier, Kasich, a Republican former presidential candidate in 2016, doubled down on his criticism in an interview with Dana Bash on CNN's "State of the Union."

"The total number of dollars that are going to be dedicated to Medicaid are not enough," Kasich said. "It's not enough resources there, and I've been very concerned in my state about treating the mentally ill, the drug addicted, the chronically ill, particularly under Medicaid expansion."