There's a rule about lawns - if, on inspection, its more weeds than lawn, then the advice is to take up what's there and re-turf. The trouble is, for you, the bindweed; you will be absolutely right that the landscapers managed to distribute pieces of root across the whole plot and then turf on top (I'm currently in the process of preventing that happening in a client's garden, because that's what the people they're using to do the works, not being able to recognise bindweed and intending to rotovate the area, will also manage to do). And that problem will still be a problem even if you lay new turf - the bindweed will continue to appear. This isn't too bad if the area is regularly mown - it won't get rid of the bindweed, it will spread underground, but at least its being cut regularly.
I'm somewhat puzzled as to why a designer (if you had one on the landscape team) would clad a slope that's too steep to be cut with grass though - is it not possible to use a hover mower on the area, even with an extended handle arrangement? If its too steep and long even for that, then some terracing would have been a better idea - and still is a better idea.
You've also mentioned briars as well as brambles - neither of these will respond to ordinary lawn or other weedkillers, but will need a brushwood killer applied to the woody roots. Brushwood killers, though, kill soft green foliage and if spilt on the soil, stop anything else growing for up to 6 months, so caution is required when using them.
Some photographs would be useful, including one of the stony area. If the grassed slope is too steep for you to walk down and dig out weed grasses and other undesirable plants, then the most sensible (and probably most expensive) option is to redesign the area with terracing. Failing that, taking the lawn up and using a wild meadow mix yearly, but even that needs cutting at certain points in the year, and reseeding from time to time. And the bindweed will make a massive contribution to the 'wildness' because its not being cut down all the time.
One other question - what kind of barrier is there between the tenanted, weed growing house next door that enables the spread of non airborne seeds from weeds into your own garden? Dandelions spread on the wind, but brambles don't.