You may like him personally, but it doesn't sound like he knows what he's doing. Putting mulch over "nightmarish" soil won't work. I'd have him remove everything he's added, then clear the land of any and all debris (glass, naiks etc.) Then look at the soil. If it is bad, start digging in as much organic matter as possible, until the soil can be easily dug. Only then, think about plantings. Find out where he got the awful soil and be sure not to use that source for anything.

Lordy, lordy!!! No need to make a big fuss!! The soil the landscaper used is very likely fill dirt. It's cheap - often free - and generally will have an assortment of non-soil items included. But so can the soil around new construction, lots of topsoil and even that in established and well-maintained gardens. None of those things (nails, plastic, glass) will affect the growth of any plantings. Just pick them out when you encounter them. NBD!!

The nut sedge may be a bigger problem but I am not convinced that is what you have. Looks more like just a weedy grass. Can you excavate carefully so we can see the roots?

I get that on the plantings but unfortunately the area in question is going to eventually be largely meadow/play space for kids so having nail and glass in the top soil is problematic. Here is a pic of one I dug out..

Better to do thing right the first time then spend years trying to remedy the things left undone at the outset. I stand by my advice.

And I stand by mine. Perhaps it is the many years of experience of gardening in a variety of locations, both my own and for clients, but I have encountered all sorts of what most folks would call "undesirable" soil conditions. And other than those that involved poor drainage, all were completely workable and virtually no soil issues affected the growth of any plants. There is nothing here that needs to be done and really nothing that needs to be "remedied" or fixed! If the area is just going to be a lawn, kid's play area or a meadow, there is really no concern to have to remove any soil, remove any non-soil bits and bobs (less you see them outright on the soil surface) or somehow think it is not safe for kids to play on (they play on asphalt or concrete at school!!). Just plant whatever you intend to cover the soil surface with and don't worry. It is not an issue you need to be concerned with. A lot of fuss and bother for no reason.

btw, that is NOT nut sedge.

When we've received loads of topsoil & compost, debris was included. I've found debris in bagged compost. Raking the soil will remove those items. No biggie.