I've taken the second approach you mentioned: I own the domain, host my email with a service provider (I use Zoho Mail), and manage my DNS through Cloudflare. Here are three insights I've gained from this experience.

First, by managing your email provider and domain DNS separately, if you ever want to switch providers in the future, you'll only need to update the MX and a few TXT records. Your email address will remain the same, and others won't even notice.

Second, set up SPF, DKIM, and DMARC right from the start. Be sure to enter the recipient address for DMARC aggregate reports so you can receive daily reports showing which servers are sending emails using your domain and whether they've passed verification.

Third, even if a new domain passes all three checks, emails sent to Outlook may still end up in the spam folder at first. This is because the new domain lacks a reputation. It's not a configuration error, and it takes time to gradually build that reputation.