Everything I mentioned works for LAN services as long as you have a domain name. You shouldn't even need to point the domain name to any IP addresses to get it working. As long as you use a domain registrar that respects your privacy appropriately, you should be able to set things up with a good amount of privacy.
Yes, you can do wildcard certificates through Let's Encrypt. If you use one of the reverse proxies I mentioned, the reverse proxy will create the wildcard certificates and maintain them for you. However, you will likely need to use a DNS challenge. Doing so isn't necessarily difficult. You will likely need to generate an API key or something similar at the domain registrar or DNS service you're using. The process will likely vary depending on what DNS service/company you are using.
In terms of privacy, you are giving your identity provider insight to each of the third party services that you use. It may seem that there isn't too much of a difference between using Google's SSO vs using your Gmail address to register your third party account. However, one big distinction is that Google would be able to see often and when you use each of your third party services.
Also, it may be impossible to restrict the sharing of certain information from your identity provider with the third party service. For example, maybe you don't want to share a picture of yourself with a service, but that service uses user profile pictures or avatars. That service may ask (and require) that you give it access to your Google account's profile picture in order to authenticate using Google's SSO. You may be able to overwrite that picture, but you also may not be able to revoke the service's ability to retrieve it. If you used a "regular" local account, that Google profile picture would never be shared with the third party service if you did not upload it directly. The same is true for other information like email, first/last/full name, birthday, etc.
There are other security and operational concerns with using SSO options. With the variety of password managers available, introduction of passkeys, and increased adoption of multi-factor authentication, many of the security benefits associated with SSO aren't as prevalent as they were 10 years ago. The biggest benefit is likely the convenience that SSO still brings compared to other authentication methods.
Ultimately it's up to you to determine if these concerns are worth the benefits of using SSO (or the third party service provider at all if they require SSO). I have a feeling the common advise will be to avoid SSO unless its an identity provider that you trust (or even better - one that you host yourself) - especially if you're using unique emails/usernames along with strong and unique passwords with multi-factor authentication and/or passkeys.