kiro-cli doctor falsely reports "Kiro CLI can not run in a process it launched" on remote hosts connected via SSH

### Checks

- [x] I have searched [github.com/aws/amazon-q-developer-cli/issues](https://github.com/aws/amazon-q-developer-cli/issues?q=) and there are no duplicates of my issue
- [x] I have run `q doctor` in the affected terminal session
- [x] I have run `q restart` and replicated the issue again

### Operating system

Amazon Linux 2 x86_64, Amazon Linux 3 aarch64

### Expected behaviour

kiro-cli doctor should pass all checks when run on a remote host connected via SSH, even when Q_SET_PARENT is forwarded from the local machine.

### Actual behaviour

kiro-cli doctor fails with ✘ Kiro CLI terminal integrations: Kiro CLI can not run in a process it launched on the remote host because it detects the SSH-forwarded Q_SET_PARENT environment variable and incorrectly assumes it's running inside a kiro-cli-spawned process.

### Steps to reproduce

1. Install kiro-cli on local Mac and on a remote cloud desktop (Amazon Linux)
2. Run kiro-cli integrations install ssh on the local machine
3. Ensure remote /etc/ssh/sshd_config has AcceptEnv Q_SET_PARENT (as required by kiro-cli setup)
4. SSH into the remote host from any terminal with kiro-cli shell integrations active (which sets Q_SET_PARENT)
5. Run kiro-cli doctor on the remote host
6. Observe the false positive error about running in a process it launched

Used kiro-cli to file this issue and it's last comment is : Your kiro-cli on the cloud desktops is properly using the SSH integration from your local Mac. The only issue is the kiro-cli doctor false positive we are already filing.

### Environment

```yaml

[q-details]
version = "1.29.2"
hash = "874d4cb11bf441cda6122c6c8569b2322b7845b0"
date = "2026-04-03T21:56:10.491106Z (6d ago)"
variant = "minimal"

[system-info]
chip = "AMD EPYC 7R13 Processor"
total-cores = 4
memory = "15.27 GB"

[system-info.os.linux]
kernel_version = "5.10.251-249.983.amzn2int.x86_64"
id = "amzn"
name = "Amazon Linux"
pretty_name = "Amazon Linux 2"
version_id = "2"
version = "2"
variant = "internal"

[environment]
cwd = "/local/home/USER"
cli-path = "/local/home/USER"
os = "Linux"
shell-path = "/local/home/USER/.toolbox/tools/toolbox/1.1.3887.0/toolbox-exec"
shell-version = "5.8"
install-method = "toolbox"
in-ssh = true

[env-vars]
PATH = "/home/USER/.local/bin:/home/linuxbrew/.linuxbrew/bin:/home/linuxbrew/.linuxbrew/sbin:/home/USER/.nvm/versions/node/v24.14.0/bin:/home/USER/.toolbox/bin:/home/USER/.cargo/bin:/usr/local/bin:/usr/bin:/home/USER/bin:/usr/local/sbin:/usr/sbin:/home/USER/.local/bin"
PROCESS_LAUNCHED_BY_Q = "1"
Q_SET_PARENT_CHECK = "1"
SHELL = "/bin/zsh"
TERM = "xterm-256color"
```