Yes, by inspecting the lawns you should see if and how many weed seeds
you are bringing in. Usually, I use only the grass cut in early May,
when most plants have not seeded yet. The rest of the grass I use to
mulch the ferns, for example, (too dark under there for any seed to
emerge). Otherwise, grass makes a pretty hot compost pile, so well
mixed with brown leaves or wood chips will cook well enough to destroy
most seeds.

Your biggest problem, however, are the weedkillers. Those are being
definitely brought in. Presumably, they will degrade faster in a hot
pile of mulch, and they won't bother the plants as much if the leaves
are kept clean of clippings. That is why I always steal the neighbors
bags of yard clippings and leaves, but never the grass bags. If you see
some of the plants declining, you will know what it is.