Who says you need to dive only with a certain affiliated agency? You can dive with whoever you want. The original premise that started this discussion was wrong. You don’t need to have 24 (or whatever) dives done with SSI for them to count.
While I meet the 24 “verified dive” minimum many times over, I’ve only ever done a max of 12 dives with anyone affiliated with SSI. If the requirement is 24 dives, but SSI only has a record of the 4 checkout dives you did with Open Water. That would seem to not be enough, right? Worst case, the instructor may ask to see your log. If you have 30 dives logged since OW, you are in. Your “verified dive” number is now 34. And when you complete the course, any checkout dives will be added in, so you might be at 38 or so “verified dives” upon completion.