The problem with that theory is that the special interests that provide said funds to Democrats do not want to enact single payer, they oppose it because single payer is less profitable for their industry. So, these Democrats have to pretend they support it. If any Democrat actually supported single payer, the pharmaceutical industry wouldn't finance their campaigns, the insurance industry wouldn't finance their campaign.
The only way to get a Democratic politician to actually support a single payer initiative is to ensure they don't accept funds from special interests and their campaign is publicly funded by citizens