Function Dependencies Causing Build To Fail

### Version info

**Angular:** 12.2.29

**Firebase:** 9.1.2 / 9.1.3

**AngularFire:** 7.1.0 / 7.1.1

**Other (e.g. Ionic/Cordova, Node, browser, operating system):** Windows Terminal / PowerShell

### How to reproduce these conditions

**Failing test unit, Stackblitz demonstrating the problem**

**Steps to set up and reproduce**

In any terminal / powershell, run: ng deploy. The process will run until it attempts to upload the function. The function fails to deploy.

**Sample data and security rules**

**package.json created in dist/{PROJECT_NAME}/functions**
```
"dependencies": {
"firebase-admin": "^9.11.1",
"firebase-functions": "^3.6.0",
"{PROJECT_NAME}": "file:..\\..\\.."
},
```

### Debug output

** Errors in Windows Terminal **

> Error when trying to deploy:
> Functions did not deploy properly.

** Error output from Google Cloud Builder Log **
"Step #1 - "build": Failure: (ID: XXXXXX) npm ERR! @angular-devkit/architect not accessible from {PROJECT_NAME}"

### Expected behavior

To deploy successfully. Using 7.1.0-rc.4, the dist package.json does not try to access root dependencies and deploys successfully.

### Actual behavior

Project will not deploy with cloud functions

Start with the ng deploy entry point and the package.json generated in dist/{PROJECT_NAME}/functions. Reproduce the deployment with Angular 12.2.29, Firebase 9.1.2/9.1.3, and AngularFire 7.1.0/7.1.1, then compare the generated dependencies with the working 7.1.0-rc.4 behavior. Done means Cloud Functions deploy successfully without trying to access root dependencies.

Written by the indexing model from the issue text.