The democrats are trying. They are insisting that if McConnell doesn't do what the house wants and call witnesses, that the trial will be unfair. Irony was never so blatant considering the house refused to allow Trump to have counsel during hearings.
Democrats know this whole thing is leaving a bad taste in the mouth of the electorate, and they are desperate to salvage some semblance of reputation. The only way they can do that is if the senate cooperates and allows witnesses. That will give dems the opportunity to use their media sock puppets to smear republicans.
\The end is already written on this.