The answer is that in a pre-supernova star, most of its mass is still in the form of hydrogen and helium. It is only the central core where the primordial H and He has fused to heavier elements.
This picture of onion layers is typically what you see in elementary text books. It is completely misleading in a quantitative sense. It schematically represents what's at the centre of the pre-supernova star, but in terms of the mass that is in each shell (it is obviously a 3d object) you get completely the wrong idea, because this diagram is only about 1 Earth diameter across, compared with the actual stellar radius of something like the distance between the Earth and the Sun!
Here is a more sophisticated plot taken from a paper by Fuller et al. (2015). It shows time until the supernova along the x-axis and the y-axis represents a radial assay of the chemical composition from the centre of the star to the outside. The initial total stellar mass is $12M_{odot}$. As you move leftwards towards the supernova explosion, notice how what is at the core changes - from being H dominated, to He dominated, to C/O dominated then Si and finally Fe (actually iron-peak elements). Note how much mass is contained within these core region for each stage of nuclear burning. The edge of the "helium core" encloses the central $4M_{odot}$ of the star. The subsequent heavier element cores inside the onion ring structure enclose significantly less mass, until the iron core is around $1.3M_{odot}$ just prior to the explosion. Blue shading indicates regions that are thoroughly mixed and homogenised by convection.
After the explosion, the neutron star that is produced will also have a mass of around $1.3-1.4M_{odot}$. In other words most of the rest of the star (about $10M_{odot}$ just prior to the explosion) gets blown out in the supernova. But of the $8.6M_{odot}$ that makes into the interstellar medium, well over half is still in the form of hydrogen and helium; the minority will be carbon, oxygen, neon, silicon, iron etc., and only a very small fraction of that will have been transformed (by the r-process) into elements heavier than iron and nickel.
Thus although the material injected back into the interstellar medium is enriched with heavier elements, there is still plenty of hydrogen to start a new generation of stars. It is also the case that star formation is an inefficient process, so the material from which the supernova progenitor formed will still mostly be around in the interstellar medium. The picture you should have is of a gradual enrichment with heavy elements, especially as the interstellar medium gets churned up and mixed through a variety of processes (including supernova explosions!).
EDIT: Here is an even more awesome picture. The lower plot shows the relative mass fraction of each element as a function of enclosed mass as you work your way out from a 15 solar mass star (it has shed 2 solar masses during its evolution). The really awesome thing is that it is animated, so it shows you the first few moments after the core-collapse and how things start to change. Note that the outer $5M_{odot}$ of the envelope is about half H and half He by mass prior to the core collapse. Lots of He and then O in the layers below that. The upper plot show how the density temperature and outward velocity are behaving. The image is from the website of Woosley and Heger (2007), a canonical work on the subject.
Hmm. I can't upload the animated gif. Here it is; well worth a look.
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