I'm just reading the rules after last playing 35 years ago. So maybe I'm missing something. But is there any downside to ordaining all colonist for free? Are they stuck with Missionary status so they can't become other specialists?

You can turn anyone you want into a missionary at the docks in Europe, but it would be unnecessary micromanagement in most cases. They will usually be more useful working in the colonies or learning new jobs, and you only need so many missionaries anyway.

They will not be stuck as missionaries. You can un-bless them anytime you want at your colonies, and they automatically revert back to their specialty if you put them to work inside a colony (in the fields or at the buildings).

Note that you can also bless any unit as a missionary for free in a colony of your own if you've built a church (requires population 3).

So early on, it seems it would make sense for flexibility.

Ordinary missionaries suck compared to expert missionaries, always best to use an expert.