You'll have to decide what you really want here. "Mixing them up so that there's an even distribution..." is not a randomised list, it's a list structured according to very specific and complex criteria.
A purely randomized list is quite simple to do in FileMaker if you use the random function to assign a random number to each record, you can sort your records on that field to get a random order.
My apologies, poor choice of wording. The big part I’m having trouble getting to grips with is can you get output from Filemaker to put records in groupings of a specific number? So produce a script to say “sort records by this criteria” then . . . . ? Can you create a script that will say for every X number of records write a particular field, then sort by that field to get a list of names?
Yes it is possible, might not need anything more than a replace field contents operation with a calculation that works like this:
Ceiling ( Get ( RecordNumber ) / 5 )
This would put a 1 into a field of the first 5 records, a 2 into the next five, 3 into the next five and so forth....
The functions that I am using can be looked up in FileMaker help if they are unfamiliar.