Non-flake attempt: I am trying to make it work like this:

{ pkgs ? import <nixpkgs> {} }:
pkgs.mkShell {
  buildInputs = with pkgs; [ dotnet-sdk_7 dotnet-runtime_7 ];
  DOTNET_ROOT = with pkgs; lib.makeBinPath [ dotnet-runtime_7 ];
}

But still failing.

This seems to be enough to get the project up and running:

{ pkgs ? import <nixpkgs> {} }:
pkgs.mkShell {
  buildInputs = with pkgs.dotnetCorePackages; [ sdk_7_0 ];

  DOTNET_ROOT = with pkgs; "${dotnet-sdk_7}";
}

If anyone finds this helpful, this is how I've turned that into a flake.nix:

{
  description = "Basic .NET 7 flake";

  outputs = { self, nixpkgs, flake-utils }:
    flake-utils.lib.eachDefaultSystem
      (system:
        let pkgs = nixpkgs.legacyPackages.${system}; in
        {
          devShells.default = 
            pkgs.mkShell {
              buildInputs = with pkgs.dotnetCorePackages; [ sdk_7_0 ];

              DOTNET_ROOT = with pkgs; "${dotnet-sdk_7}";
            };
        }
      );
}