Multichain: Consolidate Network Switching in E2Es to Make

### What is this about?

In an effort to make removal of the NetworkPicker from the UI (https://github.com/MetaMask/metamask-extension/issues/27836) header break less tests, we should consolidate all usages of the NetworkPicker in E2Es into one function, called everywhere, so that we'll only need to adjust that function when we remove the NetworkPicker.

### Scenario

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### Design

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### Technical Details

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### Threat Modeling Framework

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### Acceptance Criteria

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### Stakeholder review needed before the work gets merged

- [ ] 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

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Search the E2E tests for every use of the NetworkPicker and compare the existing network-switching flows. Introduce one shared switching function and update all those usages to call it, then verify the E2E suite passes and that future NetworkPicker removal requires changing only that function.

Written by the indexing model from the issue text.