Those infrastructure "investments" will almost certainly not generate the same return for pensioners as putting the money in an index fund.
Sure but pensioners care about consistency vs. gross returns. You really don't want your pension to lose a ton of value in a downturn because people are constantly drawing from it, it's a risk off investment. Bonds are also poor investments compared to an index fund from a gross return perspective, but that's not why people/funds buy them, they buy them to lower risk.
Perhaps not, but you also get the infrastructure, which is worth something.
Would you get the same net result if you put the money in an index fund and then bought the infrastructure? That's the actual comparison.
Maybe so, but I would contend it is worth considering the broader implications of those investments and the effects that new and upgraded infrastructure could have on the greater economy.
Speaking only for myself, I would be okay with a lower return if it also means we as a society have good public transit, roads that aren't more pothole than asphalt, water that doesn't have to be boiled on occasion, reliable power, modern internet, and so on.
CDPQinfra deals in Quebec are wonky at best.
Interestingly, Quebec in Canada has been loosely following this model recently and it's working great. Long video but interesting if you like infrastructure stuff:
https://www.youtube.com/watch?v=XlHqqA0onn0&t=532s
TLDR: CDPQ invests on behalf of the Quebec Pension Plan, and has been building infra on behalf of the pension plan across Quebec (and Canada) to great success comparatively.