you appear to have based your argument on an over simplified view of convection, namely, that the mechanism is driven directly by the density/temperature difference between material at the top and bottom of the column.
No. The argument is not based on this.
The simplification you mention is used, but simply to show the problem clearly.
The density versus depth graph and the statement:
"the hot rock, 3740 K, at the bottom of the mantle has a density of 5,560 kg/m³, and that the density decreases from 5,560 kg/m³ to 3,370 kg/m³ as one approaches the top of the mantle (3,370 kg/m³ is the density the cold rock, 930 K, at the top of the mantle, about 40 kms down)." is there to show that even locally, convection, if it occurs (which of course it cannot) still has denser rock, rising into lighter rock.
Enough sustained heating from below will result in convection throughout a given column of material without an overall relative change in density between top and bottom at any specific time.
This is wrong because you do not have unlimited heating. You only have enough heat (from the core) to heat the rock at the bottom of the mantle to 3740 K, where it has a density of 5,560 kg/m³ (which is heavier than the rock anywhere above it).
Like I have said above;
"the hot rock at the bottom of the mantle is heavier than the rock anywhere above it. Thus the hot rock at the bottom of the mantle will never rise, it will just sit at the bottom of the mantle, forever."
