No, you are completely wrong. Burning and mere heat transfer are very different in how they change temperature. Burning infuses ever more heat into a system.
You don't really believe that a small amount of steel would melt more steel, and then more steel, untill all the steel has melted? Would a drop of molteb streel melt a ton? Obviously not.
When molten steel meets solid steel, the two exchange temperature - the hot liquid cools, the cooler solid heats up, and they meet somewhere in between. If the solid was near melting point already and the liquid was well above melting point, then the solid might melt, too, but if the solid was cool, the liquid will much more likely solidify.
And that's the whole problem here: You can't keep steel molten for a long time, because it will meet cooler materials everywhere and thus cool down till it gets solid. It loses more heat by radiation, even without contact. So to keep steel molten for weeks on end, you need to add heat by burning stuff more or less continually for weeks on end. Could thermite do that? Short answer: No - it burns too quickly, or not at all, and it contains much too little energy per mass unit.