No that is not how professionals do it.

You don't need to get rid of the topsoil... bringing in new topsoil will just bring in new and different weeds. And of course you have a disposal issue that you mentioned.

Weeds are everywhere in every soil, in the wind, on your shoes, in bird droppings, on animal fur.

If your goal is grass or turf grass...the professionals would spray the area with an herbicide... probably glyphosate. Maybe something else, depending upon what is growing there. They may spray a few applications, one month apart. Though you could skip the herbicide part and just scalp mow the weeds for couple months.

Then they would overseed (possibly slit seed) the entire area in the fall, after labor day, with a premium grass seed mixture. (...not the bag of sun/shade mix available at the big box store)

Then follow the seeding with proper lawn care regime of proper watering, mowing high, fertilizing appropriately.

Your neighbor may have used sod, which is rather typical for warm season grasses. What kind of grass are you wanting to grow?