/status diagnostic false positive: ~/.local/bin not in PATH when using PowerShell

Bug description

The /status System Diagnostics section shows a warning:

⚠ Native installation exists but ~/.local/bin is not in your PATH.

This is a false positive. The binary at ~/.local/bin/claude resolves correctly and is the actively running process.

Evidence

$ command -v claude
claude

$ type -a claude
claude is /Users/<user>/.local/bin/claude

$ echo $PATH | tr ':' '\n' | grep .local/bin
~/.local/bin
/Users/<user>/.local/bin

Environment

• Version: 2.1.70 (native binary)
• OS: macOS (Darwin 25.3.0)
• Default shell: PowerShell (/usr/local/bin/pwsh)
• API provider: AWS Bedrock

Likely cause

The default shell is PowerShell, which adds ~/.local/bin to $PATH via the profile at ~/.config/powershell/profile.ps1. The diagnostic check likely evaluates PATH in a context where the PowerShell profile hasn't been sourced (e.g., reading process.env.PATH from the Node.js process or a non-login shell snapshot), so ~/.local/bin appears absent even though it's present in the interactive shell.

Suggested fix

Instead of a substring match on $PATH, check whether the claude binary is actually resolvable (e.g., which claude or command -v claude in the user's configured shell), or spawn a login shell to evaluate PATH.