Option 1
On my account
You hand me the technical side and don't think about it again.
Where the site runs
On my Cloudflare account. I set it up and keep it running.
Your domain
It stays registered to you, where you bought it. You change its nameservers once, and from then on I manage the DNS, including the records your company email depends on.
Who pays Cloudflare
I do. You don't need a Cloudflare account or a card on file there.
If we part ways
I export the site with its content and hand it over. You point the nameservers somewhere else.