The practice is part of British political culture and provides an element of theater to British politics.
I don’t think it is that unreasonable to ask that the president (and if he is away the VP) appear before a joint session of Congress once a week and take questions for a certain amount of time (in the UK it is half an hour). It seems to me it would not be that difficult to implement a rule (or custom) that required that an equal number of questions be asked from members of each party.
One of the benefits that I see of introducing President's Questions is that it could perhaps be implemented under Article 2 Section 3 of the Constitution, which would allow us to introduce some combative politics to Capitol Hill and allow us to replace, diminish, or get rid of the State of the Union.