Many years ago in the GW soil forum there was a long discussion about the use of plastic in the garden. The general consensus was that there was never a good time to use plastics. The closest use that generated any positive opinion was solarization where the plastic is removed after the underlying stuff is killed. Your instincts about that plastic are correct. It will not stop anything. All it will do is cause a huge headache for you or the next people.

Most weeds and grasses can be smothered. Solarizing might work as a start. That involves wetting the grass and soil and covering it with clear or black plastic and sealing the edges from air infiltration. Do that for a month of very sunny days. Hot sunny days are better because they tend to help with the heating. Pull it away when you think the underlying weeds are dead. After that most of the plants underneath should be rotting away. As a further precaution you can continue to smother anything alive with sand and mulch.

Most people start with a product like RoundUp to renovate. Spray that once, water daily to sprout all the weed seeds, wait a week, repeat the spray to kill the new weeds, wait another week, and you should be good to go.