Just a partial answer.
Within what you define to be "symmetrical polytopes" clearly is the subset of "quasiregular polytopes", i.e. those which can be described by a Coxeter-Dynkin diagram (i.e. follow some kaleidoscopical Wythoff construction), where just a single node is being ringed. In fact those are uniform by means of their Wythoff construction, and by virtue of having just a single node being ringed all edges would belong to the same equivalence class.
This moreover excludes reducible Coxeter groups, as those would require for more than one edge type. Obviously the ratio of circumradius to edge size gets increased, whenever any link mark of the diagram gets larger than 3. Therefore you are restricted here to links marked 3 throughout. Similarily that ratio increases, whenever the ringed node gets away from any of its ends (of a possibly bifurcated) diagram.
Running through the possible cases here, and considering the dimensional behaviour, as well as those exceptional cases of the Gosset figures, we have to state that your claim is completely right in the realm of quasiregular polytopes.
Whether you would be able to find symmetrical polytopes outside the quasiregular ones (still asking for convexity for sure) and whether there would be any counterexamples, I don't know.