Log shipping/mirroring/Availability Groups to a remote server is indeed fairly common.

As you correctly noted, the most common scenario for that is DR, where the database at the remote site is offline.

With Availability Groups, you can have read-only replicas, which would allow read workloads to run against the remote database.

With log shipping, you can run the restores using WITH STANDBY, and you can run read-only workloads against the standby database.

As you also note, during the restores the database would have to be inaccessible.

You could also use replication to push the data to a remote server, and the remote database would stay online.

If what you want is a fully-available read/write copy of the DB on a remote server, then SQL Server doesn't really offer a good option for that, as guaranteeing consistency of the data between sites is an issue (there is merge replication, but that's rarely a good idea).

All of these solutions have their own advantages and disadvantages, so which one is best for you will depend on your specific expectations and requirements.

A couple key questions to start, though, are these:

1) Is the remote copy of the database expected to handle a read-only or read/write workload?

2) How close to real-time is the data expected to be?

Also, could you confirm the version and edition of SQL Server you're on?

Cheers!