"So, in my dotage, I’ll be left to rot, unless I have a stash somewhere. No Medicare?"
Provider reimbursement is now so low that many are refusing to take new Medicare patients. It does a great job of controlling costs and it really shouldn't be responsible for all of the added costs to providers resulting from the insurance fiasco. However, what it costs is still what it costs and makes no real difference to the Medicare program except when it is artificially constrained by a "fund" of taxpayer contributions.
Like Social Security (oops - - another bubble burst?) the contributions do nothing to "fund" the program because all money re-entering the government sector "must" first cancel out the debt that it was created from and cannot survive to fund anything beyond that. All new spending, including Medicare and Social Security, is done with new money creation and the deficit/debt is nothing more than tracking how much currency remains in the economy/private sector.
Given that the results are the same with or without the "funds/contributions" except for the money removed from peoples' paychecks, (ditto for co-pays and out-of-pocket expenses) it makes little sense to even have them. It's just another way of convincing people that government can't "afford" their secure retirement or affordable healthcare.
The federal government is the monopoly creator of the US dollar when it spends into the private sector, and it "uncreates" those dollars whenever it collects a tax obligation (permanently) or sells a Treasury bond (temporarily with interest).