Gosh, that is a problem. I *loathe* bindweed (altho I suppose the flowers are sorta cute - as long as they're on someone else's property far far away

)
Solarizing, or smothering the area so it gets zero light for a year or so, would be your best bet (although if seeds are in the soil, they remain viable for a considerble number of years, IME). But that is obviously not an option in lawn. Religious thorough handpicking will eventually get rid of it but your daughter and her friends are going to end up very well-off - you might consider paying them less, disguised as 'fill this container and I will give you $X"

If that's too much work/expense and you live in an area where lawns stay damp and green, and don't mind a spotty looking lawn, I suppose a weed torch would be your best plan B, again doing it frequently and thoroughly.
IIRC bindweed is one of the weeds that is not overly impressed by Roundup or other herbicides, and personally I'm not sure that the risks of using the herbicides (to you, and to your soil/groundwater/plants) are really worth the level of control it would give. Everyone has their own opinions of course.
I don't suppose the bindweed is, perchance, mainly in an area that you would like to convert to a garden bed of whatever type? That would simplify your life. Remove absolutely all the roots you can find there, then you have several options but the best would be to solarize for a month or so (depending on your climate) and then put down cardboard or black plastic under heavy mulch and leave it that way til next year, when you could begin planting (making sure to promptly pull up any lingering bindweed that might surface).
Good luck,
Pat, whose bindweed problem was mainly confined to right along the front of the house when we moved in, and my way of dealing with it was to put down landscape fabric under heavy mulch, planting only a few large things thru holes in the landscape fabric, and weeding VIGOROUSLY. Five years later, I still get some seedlings around those plants and anytime I make a new planting hole, but seedlings are easily rubbed out with the fingertips within the first week or two, and the bulk of the problem is gone... whew
