We want to think about their return on investment for keeping you as an employee and allowing you to work remote.
Your proposal for remote work will most likely be a verbal one unless your manager asks for a typed proposal.
Bring up the idea and use your email writeup as a reference during the initial discussion. Then email the proposal for your follow-up.
When you’re applying for jobs and one of your “must have” benefits is remote work, then start your remote job search with remote-first companies.