There are a couple of key differences.
First of all, the republicans have been the ones fighting against gun regulations. And their current fuhrer, Stubby McBonespurs, has openly called for violence in the past (such as suggesting "second amendment people" can do something about Clinton, or telling his supporters to beat up protestors and he will "pay their legal bills".
As such, the shooting victims would (rightly) see a visit from a republican president as a visit from the person who was largely responsible for causing harm in the first place. "We are sorry we actually made the situation worse. But we are offering our thoughts and prayers. That's as good as actually doing something, isn't it?"
By comparison, Biden and the democrats have NOT fought against gun regulations. Nor have they openly supported violence.
Secondly, Trump's public appearances have shown him to be a complete and utter buffoon. (Like making racist jokes at a gathering of native american veterans, or telling inappropriate stories to a boy scout gathering.) It would be understandable that people might not want to be associated with Trump, since he is just as likely to turn what should be a somber occasion into a political event.
By comparison, Biden has no such reputation for going off the rails. (He is elderly, and his stutter does occasionally cause him to trip up, but he is more than capable of holding a short phone conversation with a grieving family.)