Okay, but how would it then be different from Bitcoin if anyone can mine it?
It would be no different in this regard. That is precisely the point: make it resemble Bitcoin as much as possible to gain public confidence. (It would likely have the anonymity feature, too, I imagine.)
Also, if your Imfcoin is to back up national currencies (like gold was backing up dollars up to a certain point), it shouldn't be used directly as a currency. Otherwise, there will be a case for a special kind of Gresham's Law, but even closer to the original idea than the interpretation I suggested a few year ago here. In short, why would the elites want the populace to mine it, and why would then the latter want to mine it?
In most cases, imfcoins wouldn't be used to transact, but would be a good place to park your savings in some situations. If you're a Thai and you suspect your government is issuing too much debt, you convert some of your Bahts into Imfcoin until the danger is over, or as a long term store of value. This is not different from the role of gold during the gold standard, or the role of dollars in many countries today.
I think I understand what you mean by Gresham's Law operating in this world. While stability holds, Gresham's Law won't operate, because there is bank interest. During the gold standard, holding gold earned no return and holders were losing more to inflation than holders of paper money deposits, while stability lasted. (For the same reason, developing countries today have much higher bank interest than dollar deposits anywhere.) Again, there would be no difference with Imfcoin.
How could it possibly give more stability to crippling economies?
In the same way as the gold standard while the state had enough gold to hold the system stable. It was only after states ran out of gold (after runs on the gold during crises of confidence) that the gold standard became a 'bad idea' according to establishment economists. Trust me, the elites would love to go back to the gold standard (with the economic profession changing tune again) if it were not for the embarrassment. Here is one good way to do it, from their point of view.