You are correct, they are not being replicated. It knows that NYPUBLIC_2 doesn't have a copy of the PF which is why it gives you it on the list when you try to add it.
To check replication for all PF's the easiest method is to download EXFolders from Microsoft web site and then run a replication report.
That will dump out a CSV which will tell you exactly which servers any public folder is replicated on. It could be that many of your folders are and some aren't.
Because new public folders can be added and permissioned through Outlook but only replicated via the Exchange console or powershell it is too easy to forget about adding the replicas.
If you want to replicate all public folders to another database for safety you can either set the top-level of each public folder to go to the other server and then using EXFolders force that replication entry down to all the other public folders, or there is a script in the Exchange server scripts folder that will do it all for you.
AddReplicaToPFRecursive.ps1 I think.
Do note however that if you have a user who is in a mailbox database on NYMB02 and that database points to NYPUBLIC_2 for its public folders that when you kick off replication from NYPUBLIC all their PF entries might appear to 'disappear'.
NYPUBLIC_2 has an empty copy of the PF to start with once the hierarchy has been updated and this gets filled up, eventually, by the replication process, it isn't clever enough to hand off the user to the other server until that process is complete.
If you have a big PF structure you might want to target larger folders for weekends / quiet periods etc. There is unfortunately no easy way of telling how far Exchange is through the replication process or speeding it up.
Maybe someone else can put the above more clearly.
Neill