Lots of people like and recommend Bitwarden. I think followed by KeePass on second place.
I self-host stuff because I can, because I learn something while doing it and it gives me control. And I’m running that server anyways, so I might as well install one more service on it. If you don’t want to spend your time managing and maintaining servers and services, go for the official (paid) service. That’ll do, too.
If you’re worried about your internet connection going down, either use a VPS in a datacenter or just use software that syncs to your devices. I think Bitwarden does that, your passwords will be available without an internet connection to your server. They just won’t get synced until the server is reachable again.
Sure. I think there are some areas where self-hosting is kinda mandatory because other solutions don’t fulfill my requirements. But I don’t think a password manager is part of that. It stores the passwords encrypted in the cloud anyways, $0-$10 a year isn’t much and I think Bitwarden has a good track record and you’ll be supporting them. Self-hosting is a nice hobby and I think integral part of a free and democratic culture on the internet. But it doesn’t have to be every tiny tool and everyone. Do it if you like, otherwise it’s fine if you support open source projects by paying a fair price if you want convenience and they offer a good hosted service.