Can migrate assistant transfer individual files?

Nope, MA is an all or nothing proposition. You can tell it to migrate:

• Settings
• Apps
• Data

However that is about it. If you want a clean migration then clean up the old Mac, backup using Time Machine and follow the steps in Setup Assistant and Migration Assistant for a dead reliable and fastest possible migration.

Why not use iCloud or an external drive and do copy/paste?

My Migration Assistant on my new MacBook Pro freezes at various stages during my attempt to migrate from my old iMac using the Time Machine backup on a 4 Tb USB external HD (last backup last evening). One time it transferred most data over several hours but then froze during setup with '13 minutes to go'. Today it freezes when I attempt to even start a new MA process to complete the transfer. My new MacBook is useless until I can complete this. Help!