All that strimmed green waste would rot down nicely under the weed suppressant over Winter, don't waste it. Systematic uncovering and digging over sounds good to me, whichever method you use to dig, mechanical or by hand. Uncovered ground soon repopulates with weeds, weedkiller or no.
I wouldn't let glyphosate within a million miles of my plot if I had the power that it's manufacturers do. There is possibly only one reason I would consider using something like glyphosate, and that is to deal with a stone wall being made dangerous by deeply rooted ivy, valerian or buddleia type plants.
One other thing, consider asking your nearby neighbours if they have any objections as there are immediate vicinity considerations.