This is contrary to my experience.

The number one pitfall is that your server IPs are blacklisted (directly or by association). If you use the cheapest VPS providers they will mostly be completely blacklisted since spammers have previously used them.

We use our own email servers with our own IPs for automated emails (password resets, login tokens, welcome email, etc.) and MailChimp for newsletters. We have never had any real problem with Gmail or any other email provider.

We do see some of our clients having those problems, as far as we can tell, it is mostly caused by some sort of bad rating on IP blocks.

I mainly send from my own mail server, too. If I worry that email to a cold gmail contact will end in spam, I use a simple fallback strategy. I have gmail setup to send as an alternate email address, so it appears to come from my server's domain. The contact will certainly see gmail to gmail email, and if they reply, my email address goes into their (effective) contact list.