>If we double productivity per worker, we have twice as much wealth on average.
That's not true. There are other factors at play such as demand.
If we make the average IT worker twice as productive, that doesn't mean now every IT worker is being paid twice as much, because most users aren't going to care if there are twice as many options on the app store, or twice as many bug fixes per release.
Consumption in a society will always be roughly equal to production.
There are differences due to import/export balance, investments, government borrowing etc, but as a first approximation, if GDP increases by 10%, consumption will rise by a similar amount.
About your IT worker example: Let's say s/he produces $150k/year in value and is paid $140k. If AI makes them produce $300k of value, they may not automatically get a raise. But it becomes very attractive for another employer to hire them for $200k or $250k, or even $280k.
In the medium/long term, I don't see why wages would keep proportional to produced value.