Thinking about this problem in terms of taxation and welfare is something of a red herring; the problems occur regardless of the fiscal structure. Retirement is not about redistributing money but about redistributing production. Or, as user3067860 pointed out in the comments, sometimes output of production.
For everything a retired (or a working) person buys, somebody has to put in a corresponding number of hours of work, to produce it, distribute it, etc. When you have fewer working and more retired people, you have fewer hours of work available. Worse, you need most of those hours of work at nearly the same time the demand is there (you can't bake bread 20 years in advance!)
Importantly, this is true regardless of how the retirees pay for goods or services.
The underlying mechanism how this comes about varies. If you are using a US/German-style Social Security system, you simply have too few people paying in to pay out benefits.
If you rely on stock investments, you have too few working people buying stocks and too many retired people selling stocks, leading to a drop in stock values at the exact time the stocks are needed.
If you rely on plain savings, the same problem would come about through inflation.
Thus, the demographics problem cannot be addressed by shifting how people provide for their retirements.
There are several approaches to solving this issue, but each one of them tends to have problems of its own. That's why the demographic pyramid leads to such passionate political debates.
- Flip the demographic pyramid back.
This can be done through increasing the birth rate, or through immigration. Increasing the birth rate is notoriously difficult, has a 20-year lag time, and many people feel that it has undesirable environmental consequences. Immigration tends to be politically problematic, and can negatively affect the source countries (brain drain).
- Importing goods from other countries.
This is something most of the affected countries are doing on a large scale right now, but it only works due to huge wealth disparities between countries.
This would usually mean, more automation. Aside from the increased energy use this would usually entail, this is probably the most long-term sustainable approach.
- Accepting the problem, and living with it.
This is of course not a real solution. It means returning to a society with widespread poverty in old age.
- Expecting young family members to provide for retired people.
Another approach we are currently taking. It reduces the productivity of young family members dramatically, and really shifts the burden from all young people to a relatively few of them.
Update: one important approach I forgot:
- Increase retirement age. The challenge here is that this is often not possible for other reasons. In positions that involve hard labor (such as construction work), the human body often doesn't support a later retirement age. In other professions, we mandate early retirement for safety reasons (for instance, we do not allow pilots to fly past a certain age).