They are no different in Europe than anywhere else. It is true in the animal world it is hard to find a species that's completely innocent. Bluebirds will also kill and take over a nest if they have too. Barn swallows can do the same. I can give you a link with a story about swallows sealing a nest that a sparrow took over, and they sealed it with a sparrow inside (but the link is not in English). Swifts in Europe often take over a nest by killing the birds with its claws. The american cowbird is an angel compared to the European cuckoo bird :) I could continue with these kind of stories forever. Any way, I don't have a problem with your or anybody else's advise, but you should list all the possible solutions, not just YOUR favorite.

May is the third worst month to seed northern grasses in your area. Only July and June are worse. So that is part of the reason you have so many weeds. To someone's credit, though, I would have expected far more weeds than grass by this time of year. You're doing something right!! The garden center that sold you the seed in May should be off your list of future purchases, because they don't give good advice. Based on that I also question the quality of their recommended seed. You have plenty of time to overseed this year. Fall is the time to seed. New York is a big state. If you live in the north then start with the herbicide now to seed in early Sept. If you live in the south then you can wait a week or two and seed later in Sept. Seeding early in the fall gives the new grass time to come in and for you to evaluate whether you need to thicken it up with more grass seed. Go back and reread gardengal's first post in this topic. She has a very good way of explaining those concepts. There are many Weed-b-Gon products. "Basic" WBG is a 2,4-d product, at least in my neighborhood. A slightly different product is Weed-b-Gon Chickweed, Clover, and Oxalis Killer. That one has Triclopyr (longer wait period). Formulations for these products may vary from state to state as different regulations apply. I would suggest you read more here on this forum to see what you can learn about lawn care. You can't rely on TV ads to tell you when to do things. Here are some general concepts that have become common knowledge on this and other lawn forums. New seed (northern grasses) should be applied in the fall. Weed n Feed products rarely work as advertised. Spot spray herbicides work better. Deep and infrequent watering is the way to go. Frequent watering leads to shallow roots and general brownish appearance. Spring fertilizer should go down late, not early. Wait until May or late May. Fall winterizer goes down after the grass has stopped growing but before total dormancy (before you have regular freezes). For most this is around the end of November. Mulch mow always. The clipped grass returns valuable organic material to the soil. Mow most grasses at your mower's highest setting. This leads to deeper rooting. Deep roots take advantage of moisture and nutrients deeper in the soil.

I intentionally planted this as well and seriously regret it. However, I have planted other things that are invasive and don't respond to round up. They are a bigger issue. This does respond to round up. I know this is not always the acceptable answer for those that do not like chemicals, but round up does kill it. Even with round up it takes years. Use round up on a paint brush with latex gloves and hand paint the leaves carefully.

There's a grass called Kyllinga which I've never seen personally. Compare with images of that.