There's an option for you: behind your smart splitter, put several standard splitters in a chain. Merge together all their outputs except one output from the last splitter, send the merged output to your storage and send the one output back to the sorting line. Until your storage gets full, it will send most sorted items to the storage and only small portion (depending on how you chain the splitters, it may be about 1% if you use 4) of items back to the sorting line. When your storage fills up, all the excess will reach this overflow line.
Yes it's bulky and could be solved in a more elegant way if we had splitters with priority but that's what we have available right now. It's still not perfect. If you need to rely on "something with super many containers", you could build this "super many" right for your sorting line.