Drain outside basement door
These work fine "most" of the time...but you're right, they don't take to "snaking" as they don't go very far--and they're not the greatest if there's MAJOR rainfall. It could be that your house uses one of these types of sumps. I think a house we owned many years ago had one of these. (It's not what you want to hear--but we kept a shop-vac handy when the rain wouldn't quit.)
Hopefully the rain will stop soon and give you a break!
Good luck, and hope for sun!