[dotnet-watch] The console doesn't display the final line of output unless the output ends with a newline character.

### Describe the bug
I have a multiple-lines log information to show in console. When I ran the project with `dotnet watch run` command, the final line of the output didn't display in console but displayed along with the next log output. However, the same log information can display correctly when I run with `dotnet run` command.

### To Reproduce
Here's the demo repository to show the issue: [DotNetWatchRunIssue](https://github.com/RickyLin/DotNetWatchRunIssue)

When running with `dotnet watch run` command then request the API, the demo project shows without the fina line `DemoKey: DemoValue`:
```
2025-07-17 11:14:50 [ERR WeatherForecastController ] An error occurred while fetching weather data
System.Exception: This is a test exception
```

Then request the API again, it shows:
```
2025-07-17 11:14:50 [ERR WeatherForecastController ] An error occurred while fetching weather data
System.Exception: This is a test exception
DemoKey: DemoValue2025-07-17 11:14:55 [ERR WeatherForecastController ] An error occurred while fetching weather data
System.Exception: This is a test exception
```

### Exceptions (if any)
_No Exceptions_

### Further technical details
#### dotnet --info
```
.NET SDK:
Version: 9.0.302
Commit: bb2550b9af
Workload version: 9.0.300-manifests.183aaee6
MSBuild version: 17.14.13+65391c53b

Runtime Environment:
OS Name: Windows
OS Version: 10.0.26100
OS Platform: Windows
RID: win-x64
Base Path: C:\Program Files\dotnet\sdk\9.0.302\

.NET workloads installed:
[aspire]
Installation Source: VS 17.14.36310.24
Manifest Version: 8.2.2/8.0.100
Manifest Path: C:\Program Files\dotnet\sdk-manifests\8.0.100\microsoft.net.sdk.aspire\8.2.2\WorkloadManifest.json
Install Type: Msi

Configured to use loose manifests when installing new manifests.

Host:
Version: 9.0.7
Architecture: x64
Commit: 3c298d9f00

.NET SDKs installed:
5.0.408 [C:\Program Files\dotnet\sdk]
9.0.302 [C:\Program Files\dotnet\sdk]

.NET runtimes installed:
Microsoft.AspNetCore.App 5.0.17 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 6.0.30 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 6.0.36 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 7.0.19 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 7.0.20 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 8.0.5 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 8.0.11 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 8.0.16 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 8.0.18 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 9.0.5 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.AspNetCore.App 9.0.7 [C:\Program Files\dotnet\shared\Microsoft.AspNetCore.App]
Microsoft.NETCore.App 5.0.17 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 6.0.30 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 6.0.36 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 7.0.19 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 7.0.20 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 8.0.5 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 8.0.11 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 8.0.16 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 8.0.18 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 9.0.5 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.NETCore.App 9.0.7 [C:\Program Files\dotnet\shared\Microsoft.NETCore.App]
Microsoft.WindowsDesktop.App 5.0.17 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 6.0.30 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 6.0.36 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 7.0.19 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 7.0.20 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 8.0.5 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 8.0.11 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 8.0.16 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 8.0.18 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 9.0.5 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]
Microsoft.WindowsDesktop.App 9.0.7 [C:\Program Files\dotnet\shared\Microsoft.WindowsDesktop.App]

Other architectures found:
x86 [C:\Program Files (x86)\dotnet]
registered at [HKLM\SOFTWARE\dotnet\Setup\InstalledVersions\x86\InstallLocation]

Environment variables:
Not set

global.json file:
Not found
```

#### The IDE (VS / VS Code/ VS4Mac) you're running on, and its version
- Visual Studio 2022 Version 17.14.9 (July 2025)
- Visual Studio Code 1.102.0