If the CO2 component cannot be separated out and quantified then there is no basis for saying it is a contributing factor.
There is
physical basis for saying that it is a contributing factor, which is why the current warming was predicted. Physics predicted it. It didn't precisely quantify it, of course, but it got within the ball-park.
Qualitative assessments are useful in only the simplest examples and even then are of little use in doing other than explaining a physical principle to the student.
The laws of physics are all about the quality.
Without the math physics is of no value.
Numbers aren't math. Extracting
exactly what contribution enhanced CO
2 has made to day-to-day climate change (or even decade-to-decade) is not what confirms the laws of physics - those laws which were used to
predict the current warming. Not to predict the exact warming, nor its exact impact on climate from place to place and year to year, but to predict that it would happen. Before the event.
It is not credible that global climate can change because of modest industrialization over 2-3% of the world.
Europe, Russia, North America, India, China, Japan, and other parts with slightly less intensity is a bit more than 2-3% of the land-surface. And from those parts the aerosols (being airborne) travelled to yet more distant parts, even over the oeans.
It is very credible.
I am unaware of any quantitative determination of this as fact.
The quality is good though. As explanations for what happened go, it does very well. It wasn't predicted before the event but it does go a long way to why it happened.
Again without math there is nothing which can be said on causes. That is the way it is even for simple systems. The earth's climate is rather beyond a simple system.
Far from simple, but not mysterious. By demanding exact numbers for everything you treat climate as if it
is a simple system.
It's a system which is going in one way only - ever warmer. Just as expected.