Cutting tree roots
Cutting roots will hurt the tree. Will it kill it, probably not. Will it resolve the problem probably not; I suspect the tree will just send out new roots from the cut and those roots will ultimately cause the same damage. Planting a replacement a greater distance away and at some point removing the river birch causing the patio brick damage may be a possible alternative fix. I am a bit worried that the soils may not be the best in the area; they may be tight clay soils or compacted creating little air and water space in the soil profile resulting roots on the surface in an effort to get air and possibly water.