If you go the forwarding plus send mail as route, sort out SPF on the domain or your replies will land in spam. When Gmail is doing the sending, the SPF record has to include Google, and your registrar's forwarding help page normally tells you what to add for their side.
Before using it with customers, send a test to a Gmail and an Outlook address you own and check the message headers show SPF pass.
Do you want a free address, or your own domain for free? Those are different questions.
For a plain free address, Gmail is fine. Make it a separate account from your personal one so you can hand it over later if you ever need to.
If you own a domain, the cheapest proper setup I know is forwarding mail for you@yourdomain to that Gmail (most registrars do forwarding free, and Cloudflare does too if your DNS is there), then adding the address in Gmail's "send mail as" so replies come from the domain. Zoho also had a free custom domain plan last time I looked, but it's limited and the free tier doesn't do IMAP, so you're stuck with their app.
Once the business makes real money, pay for Workspace or similar. It's not much a month.
