Terminal env vars (`WT_SESSION`, etc) aren't set when set as the Default Terminal

### Windows Terminal version

1.12.10983.0

### Windows build number

10.0.22000.0

### Other Software

_No response_

### Steps to reproduce

Change settings to launch terminal instead of command shell.
Close instances of terminal
Run powershell/pwsh
Check $env:WT_PROFILE_ID
See that it is blank

### Expected Behavior

$ENV:WT_PROFILE_ID should never be blank, because having a blank profile ID disables changing settings for that profile.

### Actual Behavior

$ENV:WT_PROFILE_ID is not set when terminal is launched with pwsh/powershell, after changing system settings.

Reproduce the issue by setting Windows Terminal as the default terminal, closing existing instances, launching powershell or pwsh, and checking $env:WT_PROFILE_ID. Trace the default-terminal launch path and environment propagation; done means WT_PROFILE_ID is populated and profile settings can be changed.

Written by the indexing model from the issue text.