Default profile can point to one deleted in state.json

### Windows Terminal version

1.17.2982.0

### Windows build number

25236.1010

### Other Software

PowerShell (potentially caused by multiple versions installed?)

### Steps to reproduce

1. Open Windows Terminal (see PowerShell Core profile)
2. Open new tab (different PowerShell profile appears)

### Helpful info
[settings.zip](https://github.com/microsoft/terminal/files/10057922/settings.zip)

`defaultProfile` is set to a legacy version of powershell core. However, that profile doesn't exist anymore. It's interesting that...
1. resolving it on startup results in a different profile than when hitting the new tab button
2. I can't seem to edit that profile? It doesn't seem like it's `hidden`. Idk why it still resolves correctly?

### Expected Behavior

Both tabs should contain the same profile

### Actual Behavior

Both tabs have different profiles.

Start with the attached settings.zip and reproduce the behavior using the reported Windows Terminal and PowerShell versions. Trace how defaultProfile is resolved at startup versus when opening a new tab, then verify that both paths select the same existing profile when the configured profile is absent from state.json.

Written by the indexing model from the issue text.