New construction help

Hello, I'm in zone 6B and recently purchased a new construction home, my first home. I'm looking for some advice as to how to get the most out of my new lawn. My front yard was seeded in the winter and my backyard was seeded about a month ago. It was seeded by hand by a landscaper hired by my builder. My concerns that I'm looking for advice on are as follows:

-Soil quality. I'm assuming my builder used cheap topsoil when constructing my home. The soil has many small pieces of glass, rocks/stones, and twigs in it, and is very sandy in places. Initially I wanted to purchase new loam, but with the size of the yard (14k+ sq feet) it was far too expensive. Deciding to stick with what I had, I tried adding a thin layer of soil from bags I bought at my local home improvement store, but it turned out to be far too much work and money to do it myself. Is this strategy even worthwhile? If so I'm contemplating purchasing soil from a landscaper in bulk and having it professionally spread.

-Excess material in lawn. As I said, many rocks, sticks, and glass in my soil. Wondering the best way to get rid of this material. Attempting to rake it up myself is an enormous task, and many of the stones are buried in the dirt from months of rain+soil being compacted from walking on it. I was thinking a power broom? But not sure if this would harm anything.

-Many patchy areas and spots with no growth. I'm not sure if I should wait more time for these spots to grow in (I'm not sure the weather has been ideal for grass growing; lots of rain, cloudy days, temps no higher than mid-60s consistently) or if I need to take action such as putting more seed down. About 2 weeks ago I reseeded the whole yard myself with a rotary spreader because I was concerned that the hand seeding that was originally done wasn't applied evenly enough. If I do need to put more seed down, do I need to do anything else, such as rake it or roll it over? Back to the adding of a thin layer of soil, would this help prevent the seed from moving? With all the rain we've had I've ended up with many clumps of seed in various spots in the yard.

Thank you for any help provided. I am new to lawn care and quite overwhelmed, just trying to enjoy a yard this summer with my family. Here are some pictures: https://imgur.com/gallery/nF3lD. Feel free to let me know if anyone needs any clarification, or if I can provide additional photos. Thanks so much!