To understand, we need to add some details:

1) For the basic density of the oceanic and continental components (but this is true even if we think about an ocean-ocean subduction zone, i.e., where one section of oceanic lithosphere subducts beneath another) we need to expand our view to the whole lithosphere. Plates represents mobile sections of lithosphere which includes crust and the upper portion of the mantle. This means that the relevant density to consider is the integrated density of the different sections of lithosphere which will reflect the individual densities of the crust and mantle components, their thicknesses, and any spatial heterogeneities in these properties. Finally, we need to consider the density of these sections of lithosphere with respect to what they are sitting on, and in the case of subduction, sinking into, i.e., the asthenosphere.

2) The integrated density of a section of oceanic lithosphere depends on its age. This is for two primary reasons, A) the oceanic crust component gets a bit denser as it ages and thus cools (as it moves away from the mid-ocean ridge at which it was produced) and B) the mantle lithosphere component effectively grows as a function of age, again also primarily as a result of cooling (in detail, it grows, implying that the oceanic section of the plate thickens, but not as a constant function of age, the growth rate asymptotes towards zero after a certain age). Taken together, this means that as a section of oceanic lithosphere ages, it gets more dense.

3) Finally, we get to the real crux of subduction, specifically that given above, eventually a portion of the oceanic lithosphere reaches a density where it is more dense than the aesthenosphere beneath it, which is unstable, allowing it to start sinking. Once an edge begins to sink, assuming the strength of the rest of the plate is sufficient to not have the tip simply break off and sink into the mantle, this "negative buoyancy force" of the subducted slab starts pulling the rest of the plate down with it as it sinks.

A simple analogy would be imagining pushing a plastic sheet out across a pool. If as the leading edge of this sheet was advancing into the pool, you were adding little weights to the end, eventually the positive buoyancy of the sheet floating on the pool surface would be overcome by the weighted edge, and this edge would start to sink. As this edge sunk, it would progressively drag down the rest of the sheet as long as that negative buoyancy force (both of the original weights, but also the added mass of the portion of the sheet now submerged) persisted and was enough to overcome the positive buoyancy of the remaining floating sheet.

Thus, ultimately, the density of continental lithosphere is relevant in the sense that it generally precludes it (i.e., continental lithosphere) from being subducted, i.e., there are limited circumstances and processes by which the positive buoyancy of continental lithosphere can be overcome for sections of it to sink (or be drug) into the mantle, but its not really relevant for understanding why some sections of oceanic lithosphere can subduct - which is really the origin of your understandable confusion. For this, the key contrast is the density of the section of subducting oceanic lithosphere and the underlying mantle, and this, hopefully is much more intuitive, i.e., dense material on top of lighter material pretty much always leads to the denser material sinking.

There are of course, many technical caveats. In reality, the sticky part of what you're asking about is how a subduction zone starts, i.e., subduction zone initiation, which is definitely probably one of the least understood parts of subduction. That's not to say we don't have good hypotheses for how this process operates, but as we'll see, they're non-unique and their remains disagreement about many of the details. Once a subduction is operating, it's definitely the negative buoyancy force of the slab, i.e., that on average, the slab is denser than the surrounding mantle, that keeps it sinking and driving sudbuction, but how does that start? Broadly, there are two mechanisms proposed, what we would call either induced or spontaneous (e.g., Stern, 2004, Stern & Gerya, 2018), which are sometimes also referred to as horizontally forced or vertically forced initiation (e.g., Shuck et al., 2022).

In short, induced (or horizontally forced) subduction initiation relies on the preexistence of compressional forces that are able to shove a tip of oceanic lithosphere into the mantle a bit (even though the density of the oceanic lithosphere might be insufficient for it to really sink). Once that happens though, the oceanic crust begins to metamorphose which causes its density to increase (e.g., by transforming into eclogites) and at the same time, release water (as a product of the metamorphic reactions occurring) that serves to "loosen" the area of the mantle which is resisting the intrusion of this small oceanic lithosphere section. Together, these can lead to stable subduction, i.e., the slab tip gets dense enough, along with decreased resisting force of the surrounding mantle, for it to start to sink and transition into a subduction zone (e.g., Soret et al., 2022).

In contrast, spontaneous (or vertically forced) subduction initiation relies instead on the density of the oceanic lithosphere to reach an unstable condition (i.e., denser than the asthenosphere/mantle) via aging and/or with the presence of an additional downward vertical force (e.g., the collapse of a mantle plume head) often in the presence of an existing weakness (e.g., an oceanic transform fault). Once started, the same processes (e.g., metamorphism of the slab, release of water, etc) from the induced example likely play a role in making the subduction zone self-sustaining.

As highlighted in the Stern papers, we have examples of both occurring and the Shuck paper highlights how the initiation of a subduction zone in one place by one mechanism (induced) can drive lateral propagation and the start of subduction elsewhere by the other (spontaneous). There are still many details we don't quite understand, so this is very much an active area of research.