When we moved into our new house, we inherited a 35-50 year-old white bird
of paradise tree (for lack of a better word). It has three main trunks, is
20 feet tall, and the trunks are all very woody and desperately need to be
cleaned up.

What should I be careful of? What should I do? What should I not do?

Take it out and put in something else. Most of the homeowners I deal with
got a white bird as part of the builder's landscape. They always look good
when they're first planted and then slowly get ugly over time. They will
spread from an underground rhizome so don't worry about a hard cutback.