That's not a problem, that is the argument. People are bad at measuring their own productivity. Just because you feel more productive with an LLM does not mean you are. We need more studies and less anecdata
The whole debate about LLMs and productivity consistently brings the "don't confuse movement with progress" warning to my mind.
But it was already a warning before LLMs because, as you wrote, people are bad at measuring productivity (among many things).
I'm afraid all you're going to get from me is anecdata, but I find a lot of it very compelling.
I talk to extremely experienced programmers whose opinions I have valued for many years before the current LLM boom who are now flying with LLMs - I trust their aggregate judgement.
Meanwhile my own https://tools.simonwillison.net/colophon collection has grown to over 120 in just a year and a half, most of which I wouldn't have built at all - and that's a relatively small portion of what I've been getting done with LLMs elsewhere.
Hard to measure productivity on a "wouldn't exist" to "does exist" scale.