It looks like you've been busy!
We had an area we couldnt work at the end of last year, and rather than letting the weeds run rampant we put carpet down.
This acted as a mulch blocking out the light, but letting water through. Come this year when we were ready to dig and plant we rolled it up and almost all weeds were dead or really weak and blanched so easy to remove. We gave it a dig and pulled out roots as we went and then planted, no worries so far.
Not tried the cardboard method, but it does sound like it makes sence.
I'm sure someone here will be able to advise.

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Simon Of Kells