I feel like I might be going insane. I SWEAR I used to be able to create a custom record with standard parent records (transaction/customer/etc.), move it from beneath the dreaded "custom" tab and into its own subtab. I can create the subtab, but cannot move the sublist to the new tab it's not there as an element to move.

perhaps answer id 32850 in suiteanswers helps?

I think @Gregory Jones is correct - going to the field on the custom record and indicating that the value is the parent seems to be the common factor on all of my custom subtabs/sublists

@Mike @Gregory Jones Thanks! That's done on them, it's the only way to get it to pop into the dreaded custom subtab in the first place.

So just the button is missing?

Yes, but also being unable to more the custom record sublist out of the custom tab is an issue.