Their retirement savings can be in public company stocks instead of government guarantees but that only works if there are people willing to buy those stocks (ex: younger people saving for retirement).
In a world with forever imploding consumer aggregate demand, there is no safe asset.
Productivity gains will also be swimming against the stream because scale advantage will deteriorate in all sectors.
And that's fine because it means consumption is lower too.
Whatever you accumulate for retirement (cash, stocks, gold, state pension points) it only represents coupons for your share of the economy of the future. Details may differ but fundamentals are the same: it a promise that the future generation will share some of their yield with you.
Denmark has private pensions like the 401k system in the US.
People can start tapping into these up to 5 years before they are 70.
In reality most people will go down in time and partially tap into private pensions and saving long before they turn 70.
This is opposed to other European countries, like Germany, that does not have these private pensions, but only retirement insurance.
Just as in Germany Denmark funds it's pensions through taxes, Germany additionally funds them through employee contributions.
In Denmark you have ratepension, which truly is an investment account and not an insurance product.
You're talking about a system of pension by capitalization, like in US, which comes with its own problems.
Namely that if you reach retirement age during a crash like in 2008, you're screwed. You have to work 5 more years hoping the market recovers in the meantime, or retire anyway and get way less than you put in.
Hopefully US people realize that this is also how US public pensions (social security) actually works.
If pensions were funded by actual retirement accounts, where employees invested money into some state fund and have it paid out for retirement this would be entirely unproblematic.
But the pension system in many EU countries is set up in a way in which tax contributions now finance pensions now. This can only work with a steadily growing economy which continually generates more revenue when more people retire. In some sense it is a literal Pyramid scheme, in which the obligations are continually pushed to younger generations, which have to contribute based on how many pensioners there are.
Especially now as EU economies become weaker this issue becomes bigger and bigger.