I think you're right that heavier elements can be made, it's just energy negative to do so. But without a nova they would never leave the inside of the star to find their way into a new planet.
But they do leave. Stars not large enough to go supernova do still form planetary nebulas when the more gradually lose their outer layers to space. Only the core is left behind to form a white dwarf. This will be the Sun's eventual fate.
No, gravitational segregation like that is a very slow process and would be overwhelmed by any convection. In Earth's atmosphere, for example, it doesn't occur until very high altitude (80 km or so) where diffusion is fast enough to overcome mixing.
"By definition, during a dredge-up, a convection zone extends all the way from the star's surface down to the layers of material that have undergone fusion."
"The third dredge-up brings helium, carbon, and the s-process products to the surface," (emphasis added)
In the early universe, stars had so little in the way of "seeds" for the s-process to act on that the few seeds that were there absorbed large numbers of neutrons, eventually producing weird stars highly enriched in lead (the end point of the s-process). These stars have been detected from lead (and bismuth) in their spectra.
s-process elements (including radioactive ones like technetium) are detected in the spectra of the stars where the process occurs, which means they are right out at the "surface".
