The burden of researching claims is on those who make the claim.
Agreed. And therefore the burden of evidence is on those asserting the historicity of Jesus, because Historical Jesus proponents are making a claim; mythical Jesus proponents are simply challenging the evidentiary basis of that claim.
2) The historicity of Pontius Pilate is closely linked to the historicity of Jesus, to say the least.
Why would that be the case? Just because PP is in the Bible?
I don't see any connection whatsoever.