For the record: I can share experience from small & mid size 2010 / 2013 farm implementations & O365 tenants only, but someone has to start :)
Document Center is collaboration place which can contain libraries where all possible things that you may need for a work with the document are tuned on by default. Versioning is enabled with 10 or more tracked versions of each document, (sometimes annoying) check-in/check-out feature is turned on, workflows can control document life cycles, document ID can be used, etc. It is designed for a use cases where large-scale document management is needed.
Based on my experience, many companies use DC on ShP 2010/2013 like intelligent file share with folder or metadata driven navigation. For sensitive large files accessible four times per year like financial reports or on the other hand for something well watched, but unchanged for a year or so, shared on public, like company policy guides. Some companies small enough to not care about IRM, file plan, retention policies and such kind of things in general, use it like hidden sub-site inside each SC for archive purposes. Data are moved there for some amount of time, before somebody shift them to backup or tape and delete forever. Personally, I do not know any customer who uses DC in SharePoint online, not for now.
Records Center is long-term archive where data are locked, due to fact that declared records by the definition cannot be modified by human or system. (Metadata can be stored and updated separately like XML files, without modifying the stored record itself). Versioning, auditing, metadata management, eDiscovery, and customisable record routing and holds that prevent deletion are enabled there. Document destruction is configured through the retention policies. Some features used there can be turned on separately somewhere else (e.g. Content Organizer, that can automatically route incoming to their proper location, based on their record type), but due the fact that document parsing functions are turned off for that site by default (property bag values promotion / demotion), it operates much faster for the file operations like upload, delete or “sent to” than ordinary site templates. It is designed like that by purpose, to be able to hold large number of data. Much more than on any other sites at all.
Based on my experience, RC was rarely used in 2010, comes better in 2013, will be much better with 2016, but still I can’t remember customer who used both in combo for one use case (e.g. store financial reports in DC for 5 years and then put them into RC remain locked), mostly because of storage ( = money, ShP/SQL often uses fast and expensive disk subsystem and customers don't want to store "dead" data there for 5 years or so). If the customer has legal department or care about legal cases worldwide (US and Europe has different set of standards per one type of document), usually have one or two RC sites in place, with few policies and content types (and what was configured at the beginning often remains unchanged for years). On the other hand, unlike DC, Record Center is used by customers with SharePoint online.
I can imagine some use cases where both are involved, but in your case little steps heading to main goal (to get all data classified, maintained and stored) are needed. One DC site template for all kind of documents sent from various portal places without proper description can be really a bad idea (not mentioning stored them like that for years). For the RC, you have to have clear classification strategy already in place, because files can be moved based on rules - type or metadata only. You need to know what type of file for which amount of time you need to store there (who is the owner if audit comes, what type of file it is, etc.) and than after configure stages, rules or moves. Usually if you can't see the additional value behind it, if it is not time saver or if it is too complicated to use, you will stop using it after some time. It happens everywhere.