It’s not the democrats that it puts pressure on, it’s the RINOs.
The democrats can be expected to vote against a Trump nominee.
It’s the RINO’s who want to vote against a Trump nominee, but know they will be seen for what they are right before an election if they do.
Several of those Democrats are in red states that went for Trump - sometimes by very wide margins. Being seen as taking marching orders from Oelosi and Schumer would not help their re-election campaigns one bit.