I've noticed gmail randomly blocks Firefox these days under the pretence of "your browser may not be secure" (i.e it doesn't persist through page refreshes), similar to how they try to make you do a captcha unless you refresh the page... I seem to have less and less control over where and how I am allowed to sign in (even thought I'm using a U2F key), and as a result I'm definitely getting pushed closer to the thresh…
You absolutely should own your own domain and use it to email somewhere besides Google. I use Fastmail but ProtonMail is great, tutanota, mailfence, etc. Getting locked out of your email is no joke you don’t want to be in that situation. I have a paid account with Fastmail and a free account with protonmail just in case something goes wrong with Fastmail I can transition my free protonmail account to paid and use it…
A custom domain is mostly a vanity measure. It does allow you to migrate to a new email service if your provider cancels your subscription, I suppose... But I'd rather only have one thing to worry about.