After President Obama visited their conference last week and took questions from them, Republicans have become very keen on institutionalizing the event the way it is in Britain, where the prime minister routinely takes questions from the opposition in Parliament.
It appears to me that Republicans are simply looking to save face from having believed their own propaganda about Obama's inability to speak without a teleprompter. Also, it's standard debating technique for underdogs to be elevated by being granted a debate with the leader. The debate makes both appear equal and the challenger has less to lose than the leader.
Nevertheless, I think Republicans are being short-sighted in their demands. Would they really have wanted the grossly inarticulate George W. Bush to have had to take questions regularly from Democrats in Congress? I think not.
Moreover, the dismal quality of presidential news conferences does not lead me to think that much useful information will be gleaned from question time with members of Congress. They will just ask "gotcha" questions or about obscure issues that are only of interest to them. Anyway, Congress already has hearings with administration officials more knowledgeable about the issues.
For these reasons, I am disinclined to think that a formal question time is any value in our system of government. However, I might be willing to support the idea if the president is allowed to ask members of Congress to answer questions from him. Once they are at risk of looking foolish I think their enthusiasm for question time will vanish rather quickly.

And Republicans don't already look foolish?
Why would Question Time be the start of their foolishness? Their foolishness has been going for at least a decade.
Except that
there's rarely anyone in the room to point out how foolish they look. They just trade the foolishness back and forth among themselves on Fox News.
This needs to happen. It may just dissolve into yet another pointless political exercise--but it would give serious people a forum to display the seriousness of their ideas relative to the shallow rhetoric of their opponents.
If the American people decide they, in fact, like the rhetoric over the ideas, then we'll know it's truly all a lost cause.
I'd love it but it couldn't possibly be,
The British "question time" is great, but the thing is British politicians are trained how to speak from their youngest days, while American politicians just can't, they live on packages press bytes ... and in the Parliamentary system all the personal and power relationships are entirely different ... the whole political culture is different.
I saw one a while back in which the Tory Leader, David Cameron, just tormented the PM, Gordon Brown, with black humor up and down that would have made John Stewart and all his writers blanch, tying various issues of the moment into how Brown had waited and suffered so long, long, long, for Tony Blair to finally keep his word and leave ... then Tony was coming right back as the prospective president of the EU, the Ghost over Gordon...
It was very funny, and the members were all whooping and laughing or hissing and booing and seemingly having a great time.
But Gordon and the Labour people all took it completely in stride, and when it was their turn they gave it all right back.
I mean, compared to one Republican here coming out with a pissant, unimaginative "you lie!", what Cameron did was a nuclear strike. But at the same time, it was all fair sport. It's how they play their game.
It is just unimaginable that the President could be mocked and laughed at to his face this way in US politics. For so many reasons:
First, literally nobody would be capable of doing it -- who in American politics has the ability to speak with such style and humor? Even the ones who can think well usually speak like they can't. They just make themselves look bad when they go off improvising. If they are smart they don't want to get into this situation at all.
Second, the President is the Head of State and so represents the U.S., the entire nation. Any disrespect shown to the President is disrespect to the nation. You just can't do it. In Britain, the Queen is Head of State -- nobody would publicly mock the Queen to her face -- while the PM is just another politician, a whole different set of rules.
Third, you know, the way everbody highly interested in US politics is so left-right contentious and self-righteous and humorless about it and devoid of calm larger perspective, any seriously sharp personal back-and-forth among the leaders in public would start a civil war.
It aint't gonna happen. The Repubs are posing for effect as always, and Obama's going to retain full control of the presidential podium and the media power it conveys.
Now some way of getting more communication going back and forth on a personal level between the party leaderships might be a good thing, one might suppose.
But if so it will have to be an "American" way, fit to our world. It's not going to be anything like what the British do.
Though I really would enjoy seeing some US politicians called out like that.
bring it on Barack...
As usual Jim Glass is spot on. Public debate, or an American version of Question Time will not match their British cousins in terms of quality (see 'The Office' or music; in particular R&B) but an open discussion, longer than the ridiculous sound bites we're used to, on the issues prevents nonsense like teleprompters and closed door meetings from happening.
But like term limits open discussion will never happen: our emperors and empresses do not want to be seen naked.
British example
It's not obvious to me that British public policy is necessarily improved by the existence of question time in their system. If anyone knows of a study on this topic please leave a reference.