The roots of the tree will continue to be at the surface no matter what you do. If you don't want the roots you will have to remove the tree.

How is the drainage in that area? When it rains do you get long standing water puddles? Or does it drain away out to the street or a ditch somewhere?

The drainage in this area is good, no standing water or puddles after the rain. It’s hard to see from the picture, but there is a slight slope towards top left of the photo

Contact your local Cornel Cooperative Extension and master gardener. There's a test you can get to find out the ph of your soil. They can tell you want to do with your soil based on the results of the test.

Or try the Logan Labs standard test, which will tell you pH and all the other information required to adjust it based on Ca:Mg:K balances, every other nutrient balance you could think of, and micro-nutrients. For $25, it's a steal.

And no, I'm not affiliated with them, or with any company that makes any of the stuff to correct your soil (I'm a systems analyst by trade). I just spend hundreds of hours a year reading them for free to keep my chemistry knowledge in practice.

Logan Labs was picked by a committee of lawn care nuts on another forum. They sent in multiple samples over a couple years looking for quality, quantity for the price, consistency testing the same sample, and reliability year after year. Logan Labs beat out UMASS in the last go-round. All the rest of the university and professional test labs fell far behind these two labs.

Your picture give few to no clues about what light conditions the site has. But I suspect it's not enough to grow grass. consider transitioning to groundcover as it, mulch or paving are the only realistic solutions when grass won't grow. If you want advice in laying out a groundcover bed, then provide a panoramic photo of the area from an important, logical point of view.

I doubt a soil test is going to do much to help grow a lawn in this area. The issue doesn't have all that much to do with soil fertility and/or pH but rather with an unsuitable environment in which to grow a lawn.

The root system of the birch is so surface oriented that there is just insufficient soil depth to get a decent lawn root system established. And you can't raise the soil depth adequately without risking the tree. Yard's groundcover bed, which would presumably include the tree as well, is a much better alternative and will require far less time and effort than trying to get grass to grow where it just cannot thrive.

Thank you all for great comments and recommendations. I know the image is not the best, and the preview on this page does not reflect the reality. If you click on the image, it will give a slightly better perspective... This is East side of the house, based on the shadows you can see from the picture, there is sun exposure, especially in early spring. We used to have a good lawn in that area, but as birch tree have matured, so did the root system and this is the result.

I was hoping to find some "root friendly" lawn advice, but sounds like there is no such thing...

"... there is sun exposure, especially in early spring." If sun is then but not during the summer, that indicates shade from tree canopy. Too much shade from trees does not a good lawn make.