Let's Encrypt has nothing to do with this problem (of Certificate Transparency logs leaking domain names).

CA/B Forum policy requires every CA to publish every issued certificate in the CT logs.

So if you want a TLS certificate that's trusted by browsers, the domain name has to be published to the world, and it doesn't matter where you got your certificate, you are going to start getting requests from automated vulnerability scanners looking to exploit poorly configured or un-updated software.

Wildcards are used to work around this, since what gets published is *.example.com instead of nas.example.com, super-secret-docs.example.com, etc — but as this article shows, there are other ways that your domain name can leak.

So yes, you should use Let's Encrypt, since paying for a cert from some other CA does nothing useful.