Update Example feature with new messenger pattern

### What is this about?

Once #14590 completes we will want to update the example feature branch/code to reflect the new patterns.

### Scenario

_No response_

### Design

_No response_

### Technical Details

_No response_

### Threat Modeling Framework

_No response_

### Acceptance Criteria

_No response_

### Stakeholder review needed before the work gets merged

- [x] Engineering (needed in most cases)
- [ ] Design
- [ ] Product
- [ ] QA (automation tests are required to pass before merging PRs but not all changes are covered by automation tests - please review if QA is needed beyond automation tests)
- [ ] Security
- [ ] Legal
- [ ] Marketing
- [ ] Management (please specify)
- [ ] Other (please specify)

### References

_No response_

First review the completion of #14590 and locate the Example feature branch/code in the metamask-mobile repository. Determine which new messenger pattern it introduces and compare the example against that implementation. Done means the example feature reflects the new pattern, but the issue provides no named files, tests, or acceptance criteria.

Written by the indexing model from the issue text.