"Feel"? I never asked one!

Respond to? Yes certainly. How important that is under normal circumstances is another matter. The effect of gravity on an electron is so much weaker than electric fields, that usually we can ignore it. Yes, experiments have in fact been done, but they are a little more elaborate than you suggest. For one thing, a wire conducting an electric current need not have any more electrons in it than otherwise; it might even have fewer.

You could however increase the number of electrons in the wire by giving it a negative charge, but that would not help much; by the time that you managed any measurable change in weight, the charge would be so enormous that something would have to give! Either the wire would come apart, or more likely the electrons would begin to boil out of the metal, or leap out in the form of sparks.

The most obvious way to achieve any precise measurement of the mass of charged particles like electrons is with a mass spectrometer, a most charming principle: look it up! For all everyday purposes, including Einsteinian relativity, the mass determines how a body of any nature whatever reacts to gravity and acceleration.

Cheers,

Jon