Macos – searching Time Machine backup data

I think you shouldn't try to do this.

I used the migration assistant to copy my files

So, you've copied your documents and some more (but not the whole backup), after you installed Mac OS X (or, onto a system that had OS X pre-installed)? If so, then the old backup is simply not a good representation of your current Mac.

If you could somehow enforce a deep traversal (for this to happen, fseventsd on your Mac must be fooled into "events log in /Volumes/.. out of sync with volume. destroying old logs.", followed by Time Machine's "Event store UUIDs don’t match [..] Node requires deep traversal"), then maybe it will become a reliable full backup after all. And then this new backup, or any newer backup, can be used for a full system restore (but not using backups that were created before the deep traversal, as I think those certainly do not match your current Mac).

If you really want to go that way, then we need more information about the problems you're currently facing. Anything in the log files (Console) for example?

I can browse the old Time Machine backup, however.

Using Finder, or using Time Machine? If, at any time, you want to restore files from any Time Machine Backup, then simply hold down Option while launching Time Machine (or while clicking the icon). This will change "Enter Time Machine" into "Browse Other Time Machine Disks".