Ashley, Tyler DeWitt is not currently active on Socratic, so I don't think that you'll get a response from Tyler himself.
To answer your question, no, he couldn't have solved the problem without knowing the density of iron. The volume of the iron can be easily calculated using the initial and final volumes of the water, as Tyler demonstrates in the video.
However, the mass of the iron can only be determined by using its density. Density establises a relationship between mass and volume for a given substance.
On exams, the problem will provide you with the density of whichever substance you're dealing with. But since this was not an exam, Tyler just showed how easily you can get this value yourself.