"dotnet sln add" emits error (yet succeeds) when adding project to an slnx (.NET 10 regression)

### Describe the bug

`dotnet sln add` emits an error message (yet succeeds) when adding a project to an slnx.

The error it emits references a project path that does not exist and is not referenced on the slnx. *But it used to*. The script that fails had recently removed the named project from the slnx. So how is it that `dotnet sln` "remembers" a prior version of the solution file in order to stumble over trying to access a project that _used to_ be referenced by the file?

### To Reproduce

```
git clone https://github.com/AArnott/Library.Template.git
cd Library.Template
git checkout issue/dotnet/sdk/51666
.\Expand-Template.ps1 -LibraryName Calc -Author Andrew
```

This error is produced (amidst other output):

```
dotnet : Invalid project `Q:\src\Library.Template\src\Library\Library.csproj`. The project file could not be loaded. Could not find a part of the path 'Q:\src\Library.Template\src\Library\Library.csproj'.
Q:\src\Library.Template\src\Library\Library.csproj.
At Q:\src\Library.Template\Expand-Template.ps1:120 char:5
+ dotnet sln add "test/$LibraryName.Tests"
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ CategoryInfo : NotSpecified: (Invalid project...Library.csproj.:String) [], RemoteException
+ FullyQualifiedErrorId : NativeCommandError
```

The line that failed is this one:

```ps1
dotnet sln add "test/$LibraryName.Tests"
```

Interestingly, this command _did succeed_ to add the new project to the solution. But it wrote an error as if it failed.

### Further technical details
details of dotnet --info

.NET SDK:
Version: 10.0.100
Commit: b0f34d51fc
Workload version: 10.0.100-manifests.355811b7
MSBuild version: 18.0.2+b0f34d51f

Runtime Environment:
OS Name: Windows
OS Version: 10.0.26200
OS Platform: Windows
RID: win-x64
Base Path: C:\Program Files\dotnet\sdk\10.0.100\

.NET workloads installed:
[wasm-tools-net6]
Installation Source: VS 18.3.11207.16
Manifest Version: 10.0.100/10.0.100
Manifest Path: C:\Program Files\dotnet\sdk-manifests\10.0.100\microsoft.net.workload.mono.toolchain.net6\10.0.100\WorkloadManifest.json
Install Type: Msi

[wasm-tools-net7]
Installation Source: VS 18.3.11207.16
Manifest Version: 10.0.100/10.0.100
Manifest Path: C:\Program Files\dotnet\sdk-manifests\10.0.100\microsoft.net.workload.mono.toolchain.net7\10.0.100\WorkloadManifest.json
Install Type: Msi

[wasm-tools-net8]
Installation Source: VS 18.3.11207.16
Manifest Version: 10.0.100/10.0.100
Manifest Path: C:\Program Files\dotnet\sdk-manifests\10.0.100\microsoft.net.workload.mono.toolchain.net8\10.0.100\WorkloadManifest.json
Install Type: Msi

[wasm-tools-net9]
Installation Source: VS 18.3.11207.16
Manifest Version: 10.0.100/10.0.100
Manifest Path: C:\Program Files\dotnet\sdk-manifests\10.0.100\microsoft.net.workload.mono.toolchain.net9\10.0.100\WorkloadManifest.json
Install Type: Msi

[wasm-tools]
Installation Source: VS 18.3.11207.16
Manifest Version: 10.0.100/10.0.100
Manifest Path: C:\Program Files\dotnet\sdk-manifests\10.0.100\microsoft.net.workload.mono.toolchain.current\10.0.100\WorkloadManifest.json
Install Type: Msi

Configured to use workload sets when installing new manifests.
No workload sets are installed. Run "dotnet workload restore" to install a workload set.

Host:
Version: 10.0.0
Architecture: x64
Commit: b0f34d51fc

.NET SDKs installed:
9.0.305 [C:\Program Files\dotnet\sdk]
9.0.306 [C:\Program Files\dotnet\sdk]
10.0.100-rc.2.25502.107 [C:\Program Files\dotnet\sdk]
10.0.100 [C:\Program Files\dotnet\sdk]

.NET runtimes installed:
Microsoft.AspNetCore.App 6.0.36 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 8.0.21 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 8.0.22 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 9.0.9 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 9.0.10 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 10.0.0-rc.2.25502.107 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 10.0.0 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.NETCore.App 6.0.36 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 8.0.21 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 8.0.22 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 9.0.9 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 9.0.10 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 10.0.0-rc.2.25502.107 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 10.0.0 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.WindowsDesktop.App 6.0.36 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 8.0.21 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 8.0.22 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 9.0.9 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 9.0.10 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 10.0.0-rc.2.25502.107 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 10.0.0 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]

Other architectures found:
x86 [C:\Program Files (x86)\dotnet]

Environment variables:
DOTNET_CLI_DISABLE_PUBLISH_AND_PACK_RELEASE [1]

global.json file:
Q:\src\Library.Template\global.json

Learn more:
https://aka.ms/dotnet/info

Download .NET:
https://aka.ms/dotnet/download