The only difference is a pension plan board/fund manager is an additional middleman that can be stupid/corrupted, hence more agency risk for the benefit recipient.
Taxpayer funded pension plans offset that agency risk with the power of being able to increase taxes, assuming their tax base is sufficient.
Increasing the size of the investing class and putting names and yearly balance statements on accounts (so voters will call bloody murder if the legislature raids the funds like they currently do with Social Security) both seem extremely useful to me.
Though, increased transparency would likely force reductions in benefits for current retirees, with current workers being the beneficiaries. Also flooding so much money into the major indexes would likely bring down investment returns, but likely resulting in a slight narrowing of income inequality.
Recently there does appear to be more noise around employee/worker benefits (including wages), hence things like the striking screen actors and auto workers (and attempted strike from railroad workers, who have shockingly terrible benefits), and renewed interest in unions.
The health of a country ought not be measured solely by its GDP and financial markets.