I haven't seen the article, but the answer here is "Yes" . . . and "No". Yes, atmospheric resistance is enough to generate temperatures well in excess of that required to melt lead, as someone already pointed out. There's a bit more involved here, and as Rich alluded to, one of those is time. It takes time for the heat to be transferred throughout the entire mass of the bullet. We're talking about milliseconds here, but it DOES take time. What we actually get is boundary layers of lead melting off, but the shape probably remains pretty much constant throughout the flight, just getting a tad smaller as it progresses downrange. The technical term for this action is "ablation."
As to cores becoming molten, yes, they absolutely do, but often in a similar manner; by boundary layers. NOT the entire core becoming liquid upon firing, but minutely, over the space of fractions of a second as the bullet progresses downrange. Eric Stecker and Berger Bullets spent some major $ to have this phenomenon analyzed some years back, and it was a very interesting series of tests. Bullets were photographed in flight with a thermal process that allowed temperature variations to bee seen as different colors on the bullet itself. The two hottest points on the bullet? The first was the bearing surface, where the rifling had engraved. This isn't really surprising, considering the frictional forces that are in process during the bullets passage through a barrel. The temps shown on the thermal images were clearly more than enough to melt lead, and they were obviously doing so. The second (and this was a bit surprising), was the meplat and very forwardmost portion of the ogive. This area was actually hotter than the engraved bearing surface, and again, was well in excess of what's needed to melt lead.
Boddington has it right, or at least, in part. It sounds like he may have expanded that notion a bit beyond what actually takes place in flight, but basically, his statement is correct.
Here's something that everyone may be able to relate to, since it's fairly common, and many shooters will have seen this; grey streaks in flight, coming from the projectile. That's melted lead, outgassing from the bullet in flight. And bullets exploding in air, with what looks like a little FLAK burst between target and rifle? Again, molten lead that outgasses violently against a jacket that has been compromised in some way. At the Small Arms Firing School at Camp Perry a couple years back, we had huge problems with this, due o a combination of thinly jacketed match bullets, fast 1x7" twist rates, and very rough throats in the high-mileage Service Rifles that the army provided for the school. As I recall, at least 10-15% of the bullets going down range would literally explode about halfway to the target, while others gave off a very pronounced smoke trails as they went downrange. The ammunition was Hornady 75 grain Match ammo. The same ammo was being used by the Marine Corps clinic being taught over on Rodriguez range, and they had no problems with it whatsoever. They also had newer (USMC) rifles with bores that were in much better condition. The next year, Rock River Arms provided brand new rifles for the school, and Hornady again provided the same ammo. Not a bit of problem, and I don't recall seeing or hearing a single complaint about bullet blow-up.
Lots of things going on here, and it sounds like Boddington oversimplified his statement. Yes, the lead melts, yes, it outgasses and departs from the bullet itself, but no, I doubt that the tips melt "completely off" as it makes its way downrange.