No medical system, public or private, has infinite money.

There will always be decisions made about which conditions get research and which don't. It's unlikely that a disease this rare would be prioritized by a purely government run system, either. There are too many more common diseases to address first.

It's a little bit the opposite. Private groups are focused on profits but there are gov programs to support the rare disease research that would otherwise go unfunded in a pure market system.