your situation is exactly what I faced last year, we took over our plots in similar condition in May.
Why the council insist on rotovating (ours was actually ploughed!) I don't know. It would be better to give you the plot as it was, and maybe waive the first year or two's fees. At least then you could use a brush cutter and turf cutting machine to remove the top layer, which would take most of the grass root with it.
If there is no actual top growth visible I wouldn't bother spraying, it's a waste, unless you intend using a long term weed killer that will prevent you growing anything for months (path clear etc).
I decided, due to the time of the year to try and get peas and beans going and then late crop cabbage, cauli and sprouts so started digging the beds for the peas. The problem is the ploughing, and harrowing had chopped the perennial weed route into tiny pieces and scattered it. I took loads out, planted peas in rows but before any pea plants were visible the grasses were knee high. My runner beans did better, they out grew the bind weed by a short neck and I had a decent crop from them.
I started a monthly regime of spraying, I'm lucky that I have a backpack sprayer - if you plan to go that route then it's well worth investing in one, however for the cost of the sprayer and all the weedkiller your going to start using you could cover the whole plot, and in the end that is what I did. I really do believe that a plot in the state you have it is the only long term solution.
I bought a 50 x 4 meter roll of black plastic from all plaz - listed as allotment cover which covered three quarters of my plot. Plans for late and winter crops were shelved and plasticopia was born. Deciding I could not do any more damage I rotovated the whole lot once the beans were finished and covered it straight after. There are downsides to long term covering, you will eventually kill the weeds, but it will take a long time, and you do risk attracting slugs who will enjoy life in the dark, damp atmosphere iunder your plastic.
Knowing what I know now, and facing the same situation again i would operate a "no bare earth" policy. Focus on crops that you can sow in pots, peat pots if needed to avoid root disturbance, or maybe look at the expanding plug system, and sow through weed membrane. Every where that is not planted cover with the cheaper plastic.
You can still enjoy decent crops of most brassicas, peas and beans, courgette etc etc but crops which favour being sown direct such as carrot etc may have to be sacrificed in the first year.
Which ever option you choose, good luck - and don't get disheartened. I started to last year when the weeds just kept coming back, but eventually the "I shall not be beaten" attitude took over, and I'm pleased to say that, having started lifting the sheeting to plant potatoes and fruit trees the soil underneath is in good condition. Not weed free yet, but much much better. I bought my local Wilko's entire stock of weed membrane last winter when it was cleared out at 75% off, so even now will plant what I can through it. Unused space will stay under cover, and as soon as a bed is empty it will be covered again, but I will give carrots, parsnips, beets etc a go this year. Hopefully I can manage what weeds do reappear.