I don't think it shouldn't be counted in government spending. It's basically funded in the same way that other government programs are funded; an income tax. To me the only real difference is that they take it before it's 'your' income, and then make it so it can't be used for other purposes instead of introduced into the general fund. However it is extremely regressive and most people don't understand it as a tax. I just wrote a paper in which a worker is given their 7.65% payroll deduction back, invests 50% in a conservative fund with a 6% return over the life of it. Not only does this beat the payments of social security by $230 a month, but it only requires 10% of the original business contribution. That drastically increases the income of the worker (it's equivalent to getting a 3.8% raise), better provides for their retirement, creates jobs, etc. All by giving people back their money. The only problem is that some people are going to just spend the money instead of saving for retirement.
I'm leery of privatizing SS for the reason you state: you can't force people to save the money. So in the end, even if the return is better for people who take advantage, taxpayers will still end up footing the bill for people who don't use the money efficiently. My point is that income is income, so if we're going to have a government program funded by income tax, then it should be the same for all income for all people. Who cares if the benefit is capped? Why should that cap the contribution? I don't get to pick and choose which programs I support with my taxes. Anyway, as I understand it, SS was designed as a social safety net, not a retirement savings. If it's a safety net, then we all have to pay into the system. If it's a retirement savings, we should have a choice. Right now, we have a hybrid that ends up just being unfair and expensive. I would like to see it vastly reworked (actually I would like to see our whole tax system reworked, but that's another topic, I suppose).
That's a good way to think about it in terms of safety net vs. retirement savings. My main problem with social security is that as people age and there are fewer workers for each beneficiary taxes will have to be raised to sustain it and benefits will be delayed to a later age. Essentially we will be paying more and retiring later. This will not stop until people stop living longer or the birth rate in the US kicks back up. Neither are going to happen so we're stuck with it.
The real problem is that it never should have reached the point where a smaller workforce makes the system incredibly unstable. Unfortunately for decades Congress saw it a a bottomless fund and kept borrowing from it, so now all that is in the SS fund is a bunch if IOUs from long dead assholes. One way to deal with it, that will probably never fly, is actually only paying out to retirees that are below the poverty line without it.