I am wondering if it is possible to limit users from being able to see A, B, C record types of one particular SObject, but only be able to create A record type? I checked the permission set settings but didn't see anything for that
This is actually straightforward
A user can create only those recordtype(s) assigned to them in the union of their profile and permission set(s)
A user can view any record, regardless of recordtype as long as they have record sharing to that record. Such sharing is determined by OWD, criteria-based sharing rules, manual sharing, apex managed sharing, territories, and ownership
From the Salesforce Security Guide
Record type assignment on a user’s profile or permission set (or permission set group) doesn’t determine whether a user can view a record with that record type. The record type assignment simply specifies that the user can use that record type when creating or editing a record.