It seems like a lot of toxicity warnings are based on an industrial scale so I'm curious about the impact of using a couple printed boxes, versus the impact of the next laziest option: synthetic landscaping "fabric" or that thick black plastic material.
When it comes to chemical pollutants and health I think it's the concentrations that are more relevant than the scale. Grams of toxins spread over a vegetable garden could easily be worse than tonnes of toxins spread over a big farm.
It's laborious to measure the amount of known toxins in your soil and even more laborious to measure the health impacts of them, so I just stick to using things that I know the soil can process naturally. Apart from not wanting to poison myself I also don't want to poison future generations or create a dystopian problem for them to have to fix.
To be honest though, in my experience mulching is a bit overrated and ends up being pretty much the same amount of work/hassle as doing a bit of a digging here and there. Also, digging, weeding and tidying help you to get to know the life of your garden and become more intimate with it, with I think is totally solarpunk :)
There's another option though. Any paint store/ department sells rolls of plain brown paper for use as a dropcloth. That's what I use instead of cardboard. It's not particularly expensive, and it's as easy to roll out as landscaping fabric.
That sounds cool :) Hopefully it only contains natural fibre. I was surprised to find out recently that toilet tissue has PFAS 'forever chemicals' in it, when I had assumed it was just paper. Even this company making natural toilet paper have had problems verifying their supply chain and manufactuting process... I'm totally on the sceptical side of anything which is currently mass produced being free of toxins but that is just me!