we cannot even remotely take claims serious that have absolutely no proof behind them

Insofar as proof = evidence, witness testimony can be direct evidence in a court of law. Barring court proceedings, testimony before Congress under oath is still evidence to be considered, even if hearsay is less definitive than other sources of evidence.

This is also a political nomination taking place in the court of public opinion, for a lifetime appointment, where its reasonable and appropriate for representatives to be persuaded by electoral pressure in light of the character testimony from such an apparently credible witness. The standards are different.

I also think that given the history of sexual assault victims staying silent, the public cost of coming forward, being ignoring, and all the evidence we have of the rarity of false allegations, that we have to give added deference to these types of allegations.

The GOP is perfectly capable of finding someone just as qualified and conservative without the alleged sexual assault and perjury baggage which has the potential to stain the congress, the GOP, and the courts for decades. They have a whole list of them.