Backend/emails: Port to host request emails to new system

[tristanlabelle]

Ports host request emails to the new email system.

I added a system to share strings between different emails. It's not the prettiest but I intend to iterate on it once I'm done porting emails.

Depends on #8715 to get the side-by-side layout for the "View request" and "Quick decline" buttons. Without it the buttons will be one on top of each other.

Testing

Inspected emails resulting from uv run dump-emails.

(See note above about stacked buttons)

Backend checklist

• Added tests for any new code or added a regression test if fixing a bug
• Run the backend locally and it works
• Added migrations if there are any database changes, rebased onto develop if necessary for linear migration history

For maintainers

• Maintainers can push commits to my branch
• Maintainers can merge this PR for me

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Project	Deployment	Actions	Updated (UTC)
couchers	Ready	Preview	May 25, 2026 12:30am

[github-actions]

📝 Release Notes

This PR does not need to be included in release notes.

Reason: This PR mainly ports existing host request emails to a new internal email system without adding a clearly new user-facing feature or fixing a major user-visible problem. Although users may notice some email layout consistency and the quick decline button remains supported, this is primarily backend/template migration work.

🤖 Bot Debug Information

Model: couchers.openai.gpt-5.4
Decision: exclude
Reasoning: This PR mainly ports existing host request emails to a new internal email system without adding a clearly new user-facing feature or fixing a major user-visible problem. Although users may notice some email layout consistency and the quick decline button remains supported, this is primarily backend/template migration work.