[AspireE2E] When `dotnet watch` on Aspire Starter App (ASP.NET Core/React) app, The command line does not get any update information, but the browser has updated automatically

REGRESSION INFO: Not a regression, new template on Aspire 13.1

**INSTALL STEPS**
1. Clean machine: Win11 x64 23h2 ENU
2. Install SDK 10.0.102
3. Install Aspire.ProjectTemplates 13.1.0
4. Install nodejs

**REPRO STEPS**
1. Open CMD and create an Aspire Starter App (ASP.NET Core/React)
```
dotnet new aspire-ts-cs-starter -o aspire01
cd aspire01/aspire01.AppHost
```
1. Run `dotnet watch`
1. After the dashboard load successfully, open the frontend app in browser.
1. Change the table title from `Weather Forecast` to `Weather` in App.tsx file of `frontend/src` folder and save the file.

**ACTUAL**
The Command line does not get any update information, but the browser has updated automatically.


**Note**:
1. This issue also reproduces on app.css file
2. This issue does not reproduce on Aspire Starter App (ASP.NET Core/Blazor) app.

**EXPECTED**
After change the file, the command line should contain the` dotnet watch` information

Start by creating the Aspire Starter App with `dotnet new aspire-ts-cs-starter`, running `dotnet watch` from `aspire01.AppHost`, and editing `frontend/src/App.tsx` or `app.css`. Compare the CLI output with the browser's automatic update; done means file changes produce the expected `dotnet watch` information as in the Blazor template.

Written by the indexing model from the issue text.