With an SSD there is a level of indirection between blocks (smallest writable amount) and pages (smallest erasable amount). A given block can be stored almost anywhere on the SSD and moves each time it is written to, both for wear leveling reasons and to avoid erasing and writing to a whole page.
Witting 1k to a SSD does not make it erase a whole page unless there are no free pages. It writes the data to one block then does whatever metadata updates required to make note that logical block X is now stored in physical location Y. Yes this means the old copy of the block could hang around for a long time - until the page it was on gets erased.
In more bare metal flash storage which does not have this level of indirection a parameter like EBS would matter but not for a SSD.