Part of your assumption is that there was no evidence to support the theory that there was some sort of passage, which at the time many believed there was. Proponents of open polar sea theory believed there was a ring of ice that once someone found a way through would lead to an open sea at the top of the world.

  Some of the evidence pointed to by proponents of these theories was the idea that the Gulf Stream in the Atlantic and the Kuro Siwo in the Pacific both fed warm water up under the ice into this sea. This was an idea that came from an American naval officer named Silas Bent who did a lot of sailing in the Pacific to come to this conclusion. There was also anecdotal evidence people pointed to of driftwood from Siberia being found in Greenland. Similarly, in Matthew Maury's book The Physical Geography of the Sea some of the evidence given is that a whale was caught in the Bering Strait with a harpoon from Greenland. These things led people to believe there had to be some sort of passage, and if there was one then whoever found it would stand to gain a lot from it, whether it be individual fame or glory or else prestige for the country that sent the expedition that found it.

  While this doesn't explain the origin of the idea of a passage, it should help explain why it took so long for people to give up on the idea.

 

I got a lot fot his information from Hampton Sides' In the Kingdom of Ice as well as general stuff I looked up while reading it. This is my first time answering here so sorry if it leaves anything lacking.