TSQL Solution to backup ALL databases on ALL instances of a server?
I've set up option 3, Master Server, a number of times, but never for scheduling backups. I'm able to schedule jobs for performance data collection, pushing out administrative changes/script updates via SQLCMD, and run tests against multiple versions of SQL Server on multiple instances to test for any unexpected changes made after SP/CU application (among other things).

The question is though: Is there a reason why you want to back up all databases on all instances from one server? Having all your backup eggs in one basket means that none of your backups get done for as long as that master server (or at least SQL Server Agent) is offline. Having backup jobs on each server is the standard for good reason (regardless of how they're administered).

Rather than writing your own backup statements, I highly recommend using Ola Hallengrens Maintenance Solution (the whole maintenance package is worth using). It just works, and works very, very well. It gives you options you won't have thought of, but will come to need in the future.

Armed with that script, you can iterate through a list of your instances, and execute MaintenanceSolution.sql (which you downloaded from Olas site) against each server. I've done this using a batch file that uses SQLCMD with a trusted connection to each server, and outputs results to a CSV for each server (assuming you've got privileges on each instance). If you use the Registered Servers view in SSMS, you can also execute that script against all servers in a given group at once.

Outside of that, you're looking at something like redgate SQL Backup Pro (I've no affiliation with them), Idera SQL Safe Backup (again, no affiliation) to do what you want, but those products do cost money outside of their respective trial periods.

Either of these solutions will give you far greater safety than scheduling backups for all databases on all servers from one instance.