The incorrect assumption is that “Conservatives” don’t want “their” check from the government.
The only actual conservative solution to Social Security is to end it completely. Stop taking it out of paychecks, stop paying out any money.
If some people can’t make it (and there will be a lot of them), then they are wards of the state based on their need.
Anything short of that is nonsense and a lie, including keeping the current system.
Government shouldn’t be making disability payments, because there is no incentive for government to turn anyone down. Bureaucrats know that someone above them will just cave and give people what they want, so why bother?
The Social Security disability program is rife with fraud.
While this might be the only way to end it completely, it's politically impossible. Everyone alive today has had payroll taxes collected from them for their entire working career, and made it difficult or even impossible for some to invest for retirement on their own.
The truth is that the inter-generational transfer of income was always doomed to fail. Progressives thought the party would never end -- population and wage growth would continue indefinitely and keep the Ponzi scheme alive. But, then reality set in.
It never occurred to them that population growth was largely because parents had lots of kids in the hope that at least one would survive accidents and disease and be sufficiently successful in life to take care of them. Take away the need for lots of kids (by providing old-age support), and the birth rate dropped like a rock. Right now, it's BELOW replacement rate in the US. Immigration is the only thing that keeps our population growing, and that's depressing wage growth.
A transition to an asset-based system like Australia and New Zealand (and Chile and Singapore) is the only long-term solution that won't be perpetually in crisis. But, the transition will be tricky: current and near-future retirees will have to accept a cut in benefits, while several generations will have to contribute payroll taxes to fund those reduced benefits without any return on investment -- all while saving for their own retirement.