This is really a long comment on the answer by Cleft accepted by the OP.

Note that there are homogeneous spaces with the fixed point property (e.g. the Hilbert cube) and there are first countable non-metrizable homogeneous spaces (e.g. Alexandroff's double arrow space). There are also homogeneous spaces (even compact ones) which are hereditarily separable but not hereditarily Lindelof and the other way around: some that are hereditarily Lindelof but not hereditarily separable. And the list goes on.

So homogeneity is far from characterizing topological groups (even in the class of compact spaces). On the other hand, a (kind of vague) question due to Kunen (I think) is: can one say something interesting about compact right topological groups that cannot be said about compact homogeneous spaces? (besides things like "do not have the fixed point property" or "admits a group operation which is continuous in one variable").

This is really a long comment on the answer by Cleft accepted by the OP.

Note that there are homogeneous spaces with the fixed point property (e.g. the Hilbert cube) and there are first countable non-metrizable homogeneous spaces (e.g. Alexandroff's double arrow space). Under additional axioms, there are also homogeneous spaces (even compact ones) which are hereditarily separable but not hereditarily Lindelof and the other way around: some that are hereditarily Lindelof but not hereditarily separable. And the list goes on.

So homogeneity is far from characterizing topological groups (even in the class of compact spaces). On the other hand, a (kind of vague) question due to Kunen (I think) is: can one say something interesting about compact right topological groups that cannot be said about compact homogeneous spaces? (besides things like "do not have the fixed point property" or "admits a group operation which is continuous in one variable").