Backfiring can mean that you have a couple of plug leads in the wrong place, which in this case I seriously doubt, or it can mean a lean mixture.

This seems to be the case however black plugs, particularly if the deposit is dry & 'sooty' are indications of a rich mixture or the wrong heat range.

At this time I'm going to guggest that it might be a lean mix caused by a low float level setting, or a blocked jet somewhere.

Since it was running 'fine' for 30 seconds before exhausting its fuel reserve I would not think that the distributor was a contributing factor in any way.

If the fuel bowl is filling with fuel, then fit the air cleaner, then swap leads on cyl numbers 2 & 3.

This will cause it to backfire & hopefully the pressure wave will dislogge any foreign material in the jet[s]
Allow it to backfire a few times then switch off. Make sure that there is no fire in the air cleaner [there shouldn't be] then re-position the leads to the correct plugs & see if that improves the situation at all.

If so, then just let it run at a fast tickover for a little while, if not, then as long as there are no other issues, it looks like a carb strip & clean.