If there is a point to Obama's recent push of talking to Republicans about a grand bargain on the budget, it doesn't have to do with striking a grand bargain on the budget. There's a chance that Obama having Republicans over for meals and budget chats will help him politically, by reinforcing his image as "the reasonable one," but even that is not all that likely.
As for how Obama's charm offensive will work as a negotiating tactic, maybe they would twenty years ago (or three years ago, when Democrats controlled the House), but Eric Cantor and his minions don't really negotiate so much as make demands. Darrell Issa (R-Cali.) had this to say after meeting with Obama:
“Well, he doesn’t want to balance the budget in 10 years, and he wants tax increases and he wants new spending. But other than that we’re close.”
Issa's quote is telling. Republicans weren't planning on balancing the budget in 10 years last year. Romney-Ryan talked about balancing the budget in 25 years. Paul Ryan's most recent budget claims to balance the budget in 10 years, and so now, that's a new bright line for the GOP. Agree to the 10 year plan or you're one of them. "Tax increases" include the "Buffett rule," so named because billionaire Warren Buffett pays a smaller percent of his income in taxes than his secretary due to capital gains being treated differently than ordinary income. If the GOP had to publicly defend this more often, it might start to hurt them, but that's not really happening these days, so they can call everything a tax increase and dig in their heels. As for "new spending," yes, we could use some of that. Fun fact: government spending helps the economy! And the economy is not fully recovered, despite the most recent encouraging news. So, yes, some much needed infrastructure improvements could go a long way toward strengthening the economy.
The above paragraph is nothing new: Obama won the 2012 election with those ideas. Most House Republicans are in safe districts, however, and they don't have to worry about losing their jobs for not negotiating with Obama. Quite the opposite, in fact: their only real fear is getting primaried from the right, and the best way to ward that off is to be against whatever Obama is for.
So, I'm sure the food was good and the atmosphere friendly at the meals Obama had with congressional Republicans, but otherwise I doubt anyone got much out of it.