It could if you let it! So don't let it. :) I spent a lot of time finding my domain and eventually found one that makes people chuckle when they read it, it's lightly funny and innocuous. I work in the tech industry, many (many) people have their own domains - it's not as uncommon as you'd think depending on which circles of friends. It's very common to see a domain based on their last (family) name and several family members share it, I see that a lot and have seen it for 2 decades.
It also means that you can move to another provider without having to change your address. I generally recommend having your own domain. It means you can get you own name (and partners/kids can get their own name too) as the local part, otherwise you're fighting for a good localpart name against every other Fastmail customer!
I think that says more about the domain you own than anything else. My last name is fortunately rare enough that my email is firstname@lastname.com, and the only comments i’ve gotten are from nontechnical people who didn’t know that was possible and are impressed. I suspect technical people understand how easy it is and just think “must be nice having the .com of your last name still be available for less than tens of thousands of dollars”
I've been using my custom domain email for years and no one really cares, except maybe for a few techy friends who have the same. Very rarely I've run into badly written email validators that only accept well-known email domains.
